DSpace at VNU: Nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material plate with metal-ceramic-metal layers on elastic foundation

The online version of this article can be found at: DOI: 10.1177/1077546313489717 published online 8 July 2013 Journal of Vibration and Control Nguyen Dinh Duc and Pham Hong Cong with m

The online version of this article can be found at:

DOI: 10.1177/1077546313489717

published online 8 July 2013

Journal of Vibration and Control

Nguyen Dinh Duc and Pham Hong Cong

with metal-ceramic-metal layers on elastic foundation Nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material plate

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Nonlinear dynamic response

of imperfect symmetric thin

sigmoid-functionally graded material

plate with metal-ceramic-metal layers

on elastic foundation

Nguyen Dinh Duc and Pham Hong Cong

Abstract

This paper presents a first proposal to investigate the nonlinear dynamic response of imperfect symmetric thin sigmoid-functionally graded material (S-FGM) plate resting on an elastic foundation and subjected to mechanical loads The formulations use classical plate theory taking into account geometrical nonlinearity, initial geometrical imperfection of the S-FGM plate and stress function The volume fractions of metal and ceramic are applied by sigmoid-law distribution (S-FGM) with metal-ceramic-metal layers The nonlinear equations are solved by the Runge-Kutta and Bubnov-Galerkin methods using stress function The obtained results show the effects of material, imperfection and elastic foundations on the dynamical response of S-FGM plate

Keywords

Classical plate theory (CPT), elastic foundation, imperfection, nonlinear dynamic response, thin S-FGM plate

1 Introduction

Functionally graded materials (FGMs), which

micro-scopically are composites and made from a mixture of

metal and ceramic constituents, have received

consider-able attention in recent years due to their high

per-formance heat resistance capacity and excellent

characteristics in comparison with conventional

com-posites Therefore, FGMs have been chosen for use in

temperature shielding structure components of aircraft,

aerospace vehicles, nuclear plants and engineering

structures in various industries As a result, many

inves-tigations have been carried out on the dynamics and

vibration of FGM plates and shells

One of the most popular FGM structures, which has

been widely studied by using a simple power-law

distri-bution (P-FGM) of the elastic modules varying

with thickness, is a metal-ceramic composite structure

An advantage of this is that the ceramic layer plays

a similar role to the thermal resistance, whereas the

metal layer will protect the mechanical deformation

Recently, the static and dynamical properties of

P-FGM have attracted the interest of the research com-munity In this paper, we narrow our study on dynam-ical properties, and therefore just summarize the significant findings of the last few years

Firstly, we should mention the findings of Zhao et al (2004), who studied the free vibration of a two-sided simply supported laminated cylindrical panel via the mesh-free kp-Ritz method This research has indeed provided us with the approximation calculation of the FGM panel In 2004, Vel and Batra also reported a three dimensional exact solution for the vibration of FGM rectangular plates Also in this year, Sofiyev and Schnack (2004) investigated the stability of functionally graded cylindrical shells under linearly

Vietnam National University, Hanoi, Vietnam

Corresponding author:

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

Email: ducnd@vnu.edu.vn Received: 13 February 2013; accepted: 15 April 2013

Journal of Vibration and Control 0(0) 1–10

! The Author(s) 2013 Reprints and permissions:

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increasing dynamic tensional loading and obtained the

result for the stability of functionally graded truncated

conical shells subjected to a periodic impulsive loading

Sofiyev et al (2005) also published the result of the

stability of P-FGM ceramic-metal cylindrical shells

under a periodic axial impulsive loading Ferreira

et al (2006) received natural frequencies of FGM

plates with a meshless method Zhao et al (2006) has

also developed the element-free kp-Ritz method (which

was applied successfully to an FGM panel in 2004) to

calculate the free vibration analysis of the complicated

FGM structures, i.e conical shell panels

It is not only the mechanical effects that have been

investigated, the studies of the dynamical properties of

P-FGM structures under the thermal and

thermo-mechanical loads have also been a particular interest

of many authors 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 rotating cylindrical shells

sub-jected to periodic axial loads Woo et al (2006)

investi-gated the nonlinear free vibration behavior of

functionally graded plates Kadoli and Ganesan (2006)

studied the buckling and free vibration analysis of

func-tionally graded cylindrical shells subjected to a

tempera-ture-specified boundary condition Wu et al (2006) also

published their results on nonlinear static and dynamic

analysis of functionally 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

spher-ical caps In 2008, Darabi et al obtained the nonlinear

analysis of dynamic stability for functionally graded

cylindrical shells under periodic axial loading

Matsunaga (2008) analyzed natural frequencies and

buckling stresses of FGM plates using 2-D

higher-order deformation theory

Moreover, the P-FGM plate and shell structures

under the complicated dynamic loads have recently

been a hot topic in the research community Shariyat

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

buckling for suddenly heated temperature-dependent

FGM cylindrical shells under combined axial

compres-sion and external pressure as well as under

thermo-electro-mechanical loads Allahverdizadeh et al

(2008) studied nonlinear free vibration and nonlinear

forced vibration for thin circular functionally graded

plates Sofiyev (2009, 2011) investigated the vibration

of freely supported FGM conical shells subjected to

external pressure and clamped FGM conical shells

under external loads Shen (2009) published a valuable

book, Functionally Graded Materials, Nonlinear

Analysis of Plates and Shells, in which the results

about nonlinear vibration of shear deformable FGM

plates are presented Zhang and Li (2010) studied the dynamic buckling of FGM truncated conical shells sub-jected to non-uniform normal impact load Ibrahim and Tawfik (2010) investigated limit-cycle oscillations

of FGM plates subjected to aerodynamic and thermal loads Mohammad and Singh (2010) studied the dynamic response of P-FGM plates using first order shear deformation theory by finite element method Fakhari and Ohadi (2011) also investigated nonlinear vibration control of P-FGM plates with piezoelectric layers in a thermal environment using the finite element method Unlike the others, they have assumed that the material properties of P-FGM are temperature-dependent Mollarazi et al (2012) presented analysis

of free vibration for FGM cylinders using a meshless method The dynamic stability of FGM skew thin plate subjected to a uniformly distributed tangential follower force has been investigated by Miao Ruan

et al (2012) Najafov et al (2012) studied the vibration

of axially compressed truncated conical shells with a functionally graded middle layer surrounded by elastic medium

In order to increase the loading ability, a very good choice is often the stiffener Therefore, the research on the dynamics of stiffened FGM plates and shells has also been of interest Recently, Bich et al (2012) inves-tigated nonlinear dynamical analysis of eccentrically stiffened functionally graded cylindrical panels using the classical shell theory Duc (2013) studied the non-linear dynamic response of imperfect eccentrically stif-fened FGM double curved shallow shells on an elastic foundation We have witnessed that dynamic analysis

of P-FGM plates and shells has comparatively received

a lot of attention over the last two decades

However, it is not only the metal-ceramic and ceramic-metal of P-FGM mentioned above that dis-plays high thermal resistance – in modern engineering and technology, there are also many structures that share similar properties In order to increase the adapt-ing ability 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 (Duc and Cong, 2012) Moreover, FGM can be used with the top and bottom metallic surfaces together with the ceramic core

to increase the strength and thermal resistance The sigmoid-FGM (S-FGM) plate considered in this paper with metal-ceramic-metal layers is an example

of these structures

This paper presents a first proposal to investigate the nonlinear dynamic response of imperfect symmetrical, thin S-FGM plates with metal-ceramic-metal layers on

an elastic foundation using classical plate theory Numerical results for the dynamic response of the S-FGM plate are obtained by Bubnov-Galerkin and Rugge-Kutta methods and using stress function

2 Nonlinear dynamics of imperfect

S-FGM plate

Consider a thin rectangular S-FGM plate that consists

of functionally graded materials and rests on an elastic

foundation The outer surface layers of the plate are

metal-rich, but the midplane layer is purely ceramic

(Figure 1) Please note that the dynamic response of

S-FGM plate with ceramic-metal-ceramic layers was

considered by Duc and Cong (2012)

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, h=2  z  h=2

The length, width and total thickness of the plate are

a, b and h, respectively (Figure 1)

By applying a simple Sigmoid-law distribution, the

volume fractions of metal and ceramic, Vmand Vc, are

assumed as:

VcðzÞ ¼

2z þ h h

, h=2  z  0

2z þ h h

, 0  z  h=2

8

>

>

>

>

and VmðzÞ ¼1  VcðzÞ:

ð1Þ

where the volume fraction index N is a non-negative

number that defines the material distribution and can

be chosen to optimize the structural response

From equation (1) the effective properties of the

S-FGM plate can be written as follows (Duc and

Cong, 2012):

E, 

½  ¼½Em, m

þ½Ecm, cm

2z þ h h

, h=2  z  0

2z þ h h

, 0  z  h=2

8

>

<

>

:

ð2Þ

where

Ecm¼EcEm, cm¼cm ð3Þ and the Poisson ratio  is assumed constant ðzÞ ¼ : Suppose that the symmetrical S-FGM plate is sub-jected to a transverse load of intensity q0 In the present study, the classical theory of thin plates is used to obtain the motion and compatibility equations, as well as expression for determining the dynamic response of the S-FGM plate

The train-displacement relations taking into account the von Karman nonlinear terms are (Brush and Almroth, 1975)

"x

"y

xy

0

@

1

A ¼

"0 x

"0 y

0xy

0

@

1

A þ z xy

2xy

0

@

1

with

"0 x

"0

0 xy

0

@

1

A ¼

u,xþw2 ,x=

v,yþw2 ,y=

u,yþv,xþw,xw,y

0

@

1

A, xy

xy

0

@

1

A ¼  ww,xx,yy

w,xy

0

@

1 A: ð5Þ where "0

xand "0are the normal strains, 0

xyis the shear strain on the midplane of the plate; u, v, and w are the midplane displacement components along the x, y, and

zaxes ð, Þ indicates a partial derivative

The strains are related in the compatibility equation

@2"0 x

@y2 þ@2"0y

@x2 @2xy0

@x@y ¼

@2w

@x@y

 2

@2w

@x2

@2w

@y2: ð6Þ Hooke law for an FGM plate is defined as

x, y

1  2ð"x, "yÞ þð"y, "xÞ

xy¼ E 2ð1 þ Þxy:

ð7Þ

Figure 1 S-FGM plate on elastic foundation

The force and moment resultants of the plate can be

expressed in terms of stress components across the plate

thickness as

Ni, Mi

Zh=2

h=2

ið1, zÞdz, i ¼ x, y, xy ð8Þ

Inserting equations (4) and (7) into equation (8) gives

the constitutive relations as

Nx, Mx

1  2hðE1, E2Þ"0xþ"0y

þðE2, E3Þ xþy

Ny, My

¼ 1

1  2hðE1, E2Þ"0yþ"0x

þðE2, E3Þ yþx

Nxy, Mxy

2 1 þ ð ÞhðE1, E2Þxy0 i

þ 1

1 þ  ðE2, E3Þxy

:

ð9Þ

where:

E1 ¼Emh þ Ecmh

N þ1, E2¼0

E3 ¼Emh3

12 þ

Ecmh3

2ðN þ 1ÞðN þ 2ÞðN þ 3Þ:

ð10Þ

For using late, the reverse relations are obtained from

equation (9)

"0x, "0y

¼ 1

E1 Nx, Ny

 Ny, Nx

xy0 ¼2 1 þ ð Þ

E1

Nxy:

ð11Þ

The equations of motion for a thin FGM plate on the

elastic foundation based on the classical plate yheory

(CPT) and can be written as (Nayfeh and Pai, 2004)

@Nx

@x þ

@Nxy

@y ¼J0

@2u

@t2J1

@3w

@x@t2

@Nxy

@x þ

@Ny

@y ¼J0

@2v

@t2J1 @

3w

@y@t2

@2Mx

@x2 þ2@

2Mxy

@x@y þ

@2My

@y2 þNx@

2w

@x2

þ2Nxy

@2w

@x@yþNy

@2w

@y2 k1w þ k2r2w þ q0

¼J0@

2w

@t2 þJ1 @

3u

@x@t2þ @3v

@y@t2

J2

@4w

@x2@t2þ @4w

@y2@t2

:

ð12Þ

where Ji¼Rh=2

h=2ziðzÞdz ði ¼0, 1, 2Þand k1is Winkler foundation modulus and k2 is the shear layer founda-tion stiffness of Pasternak model

With

J0¼mh þ cmh

N þ1; J1 ¼0

J2¼mh3

12 þ

cmh3

2ðN þ 1ÞðN þ 2ÞðN þ 3Þ:

ð13Þ

The substitution of equation (9) into equation (12) leads to:

@Nx

@x þ

@Nxy

@y ¼J0

@2u

@t2,@Nxy

@x þ

@Ny

@y ¼J0

@2v

@t2: ð14Þ D2w  Nx

@2w

@x2þ2Nxy

@2w

@x@yþNy

@2w

@y2k1w þ k2r2w

þJ0

@2w

@t2 J2

@4w

@x2@t2þ @4w

@y2@t2

¼q0:

ð15Þ

where  ¼ @2=@x2þ@2=@y2 and D ¼ E3

1v 2 For solving equations (14) and (15) we introduce stress function ’ ¼ ’ðx, yÞ so that

Nx¼@2’

@y2; Ny¼@2’

@x2; Nxy¼  @2’

@x@y ð16Þ Volmir’s assumption can be used in the dynamical ana-lysis (Volmir, 1972; Bich et al., 2012; Duc, 2013) By taking the inertia J0 2u

@t 2!0 and J0 2v

@t 2!0 into consid-eration because u  w, v  w, equations (14) are satis-fied Inserting equation (16) into the equation (15) for perfect plate leads to

D2w



@2’

@y2

@2w

@x22 @

2’

@x@y

@2w

@x@yþ

@2’

@x2

@2w

@y2 k1w þ k2r2w

0 B B

1 C

CþJ0@

2w

@t2

J2 @

4w

@x2@t2þ @4w

@y2@t2

¼q0:

ð17Þ

Equation (17) includes two dependent unknowns w and

’ To obtain a second equation, relating the unknowns, the geometrical compatibility for an imperfect plate can

be used:

"0x,yyþ"0y,xxxy,xy0 ¼w2,xyw,xxw,yy

 w,xy

 2

w,xxw,yy

in which w is a known function representing an initial

small imperfection of the S-FGM plate Setting

equa-tions (11) and (16) into equation (18) gives the

compati-bility equation of an imperfect FGM plate as

1

E1

@4’

@x4þ2 @

4’

@x2@y2þ@4’

@y4

¼ @2w

@x@y

 2

@2w

@x2

@2w

@y2

@2w

@x@y

 2

@2w

@x2

@2w

@y2

ð19Þ

For an imperfect FGM plate, equation (17) is modified

into form as (Bich et al., 2012; Duc, 2013):

D2ðw  wÞ



@2’

@y2

@2w

@x22 @

2’

@x@y

@2w

@x@y

þ@2’

@x2

@2w

@y2k1w þ k2r2w

0 B B

1 C

CþJ0

@2w

@t2

J2

@4w

@x2@t2þ @4w

@y2@t2

¼q0

ð20Þ

Equations (19) and (20) are the basic relations used to

investigate the dynamic response of imperfect S-FGM

plate on elastic foundations They are nonlinear in the

dependent unknowns w and ’

Suppose that the S-FGM plate is simply supported

at its edges and subjected to q transverse loads q0ðtÞ

The boundary conditions can be expressed as

w ¼0, Mx¼0, Nx¼0, Nxy¼0; at x ¼ 0, a

w ¼0, My¼0, Ny¼0, Nxy¼0; at y ¼ 0, b:

ð21Þ The mentioned conditions (21) can be satisfied if the

deflectionwand the stress function ’ are represented by:

w ¼ f ðtÞsin mxsin ny

’ ¼ gðtÞsin mxsin ny

wðx, yÞ ¼ f0sin mxsin ny

ð22Þ

where m¼m

a, n¼n

b

in which m, n ¼ 1, 2, , are natural numbers

repre-senting the number of half waves in the x and y

direc-tions respectively; f tð Þ is the deflection amplitude;

f0¼const, varying between 0 and 1, represents the

size of the imperfections

The introduction of equation (22) into equations (19)

and (20) and applying the Galerkin method gives

gðtÞ  4mþ22m2nþ4n ab

4 ¼ E1 f

2

ðtÞ  f20

  4mn

3 : ð23Þ

D f ðtÞ  fð 0Þ4mþ22m2nþ4n ab

4

 8mn

3 f ðtÞ gðtÞ  k1f ðtÞ

ab

4 k2f ðtÞ 

2

mþ2n

4

þJ0€fðtÞab

4 þJ2 

2

mþ2n

  €f ðtÞab

4 ¼q0

4

mn

: ð24Þ

m, n - odd numbers

Equations (23) and (24) can be simplified as follows:

€fðtÞ þ m1f ðtÞ þ m2f3ðtÞ þ m3f0 ¼m4q0: ð25Þ

in which:

J0B2

hþJ2m2B2þn2

2



4 D mð 2 B 2 þn 2Þ2

4B a B 2 128m

2 n 2 B 3 E 1 f 2

9B 2ðm 2 B 2 þn 2Þ2

þK1 DB 3

4B 2 þK2DBa

2ðm 2 B 2 þn 2Þ

4B 2

2 6 4

3 7 5

J0B2

hþJ2m2B2þn2

2

2n2B3

aE1 9B2

hm2B2þn22

m3¼  4D m 2B2aþn22

J0B2

hþJ2m2B2þn2

2

B2 h

J0B2

hþJ2m2B2þn2

2

mn 2

Ba¼b

a, Bh¼

b

h, K1¼k1

a4

D, K2¼k2

a2

D,

D ¼ D

h2, J0¼h2J0, J2¼J2, E1¼E1

h2

Equation (25), for obtaining the nonlinear dynamic response, the initial conditions are assumed as

f ð0Þ ¼ f0, f



ð0Þ ¼ 0 The applied loads varying as a function of time The nonlinear equation (25) can be solved by the Newmark’s numerical integration method

or by the Runge-Kutta method

3 Numerical results and discussion

The imperfect symmetrical S-FGM plate considered here a square plate: a ¼ b ¼ 1 m, h ¼ 0:01 m The plates are simply supported at all edges The combin-ation of materials consists of aluminum (Em¼70:109N=m2, m¼2702 kg=m3) and alumina (Ec¼380:109N=m2, c¼3800 kg=m3) The Poisson ration  is chosen to be 0:3 for simplicity The plate subjected to an uniformly distributed excited transverse load q0ðtÞ ¼ psin t

The nonlinear dynamic response of the S-FGM plate

acted on by the harmonic uniformly excited transverse

load q0ðtÞ ¼ psin t is obtained by solving equation

(25) combined with the initial conditions and by use

of the Runge-Kutta method

Figure 2 shows the graph of maximum deflection

of 38 periods; Figure 3 shows the nonlinear

response of the FGM plate of long period with

differ-ent intensity of loads: p ¼1500 N=m2 and

p ¼2500 N=m2;  ¼ 500, N ¼1 As yet, there have

been no reports on the static and dynamic buckling for

symmetric S-FGM plate with metal-ceramic-metal layers We are therefore limited in our comparisons The relation of maximum deflection and the vel-ocity of maximum deflection when ðN ¼ 1Þ and

q0ðtÞ ¼1500 sinð500tÞ are presented in Figure 4 Figure 5 shows the influence of power law indices on nonlinear dynamic responses of the S-FGM plate (N ¼ 1, 2, 3; q0ðtÞ ¼1500 sinð500tÞ)

Figure 6 shows the effect of the imperfection (f0¼0:001, 0:003) on nonlinear dynamic responses

of the S-FGM plate Figure 6 is chosen with

Figure 3 Dynamic response with different intensity of loads

Figure 2 Dynamic response of the S-FGM plate (N ¼ 1,

q0ðtÞ ¼ 1500 sinð500tÞ) Figure 4 Deflection–velocity relation

Figure 6 Influence of imperfection on nonlinear dynamic response of the S-FGM plate.

Figure 5 Nonlinear response of S-FGM plate with variety of volume fraction index N

q0ðtÞ ¼75000 sinð500tÞ and N ¼ 1 The increase in

imperfection will lead to the increase of the amplitude

of maximum deflection

Figure 7 shows the influence of elastic foundations

K1, K2 on nonlinear dynamic responses of the S-FGM

plate with q0ðtÞ ¼1500 sinð500tÞ and N ¼ 1 From

Figure 7 we conclude that these elastic foundations

have a strong effect on the nonlinear dynamic response

of the S-FGM plate Compared to the case correspond-ing to the coefficient K1, the Pasternak-type elastic foundation with the coefficient K2 has a stronger effect Figures 8 and 9 show the effect of the geometrical parameters on nonlinear dynamic response of the S-FGM plate Figure 8 shows the effect of dimension ratio b=a on the nonlinear dynamic response of the S-FGM plate with q0ðtÞ ¼1500 sinð200tÞ, N ¼ 1 Figure 8 Effect of dimension ratio b=a on nonlinear dynamic response of the S-FGM

Figure 7 Influence of elastic foundations on nonlinear dynamic responses of the S-FGM plate

Figure 9 shows the effect of dimension ratio a=h on

nonlinear dynamic response of the S-FGM plate with

q0ðtÞ ¼1500 sinð500tÞ, N ¼ 1

Figures 7–9 show us that the mechanical loading

abil-ity in S-FGM plate is better than P-FGM plate under

the same conditions, i.e the size and the external force

4 Conclusions

This paper presents the first proposal to investigate the

nonlinear dynamic response of imperfect symmetric

thin S-FGM plate with metal-ceramic-metal layers

rest-ing on an elastic foundation Numerical results for the

dynamic response of the S-FGM plate are obtained by

Rugge-Kutta method and stress function The obtained

results show the effects of material, imperfection, elastic

foundations and geometrical parameters on the

dynam-ical response of S-FGM plates Thus it is obvious that

the dynamic response of the considered S-FGM plate

depends on many factors significantly: volume ratio N,

elastic foundation, imperfection and geometrical

par-ameters of the FGM plate Therefore, when we

change these parameters, we can actively control the

dynamic response of the S-FGM plate

Funding

This work was supported by Project QGDA.12.03 of the

Foundation for Science and Technology Development of

Vietnam National University, Hanoi The authors are

grate-ful for this support

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Figure 9 Effect of dimension ratio a=h on nonlinear dynamic response of the S-FGM