A higher-order hyperbolic shear deformation plate model for analysis of functionally graded...

A higher-order hyperbolic shear deformation plate modelfor analysis of functionally graded materials

Trung-Kien Nguyen

Received: 11 December 2013 / Accepted: 10 May 2014

� Springer Science+Business Media Dordrecht 2014

Abstract This paper presents a new higher-order

hyperbolic shear deformation theory for analysis of

functionally graded plates. In this theory, the trans-

verse shear stresses account for a hyperbolic distribu-

tion and satisfy the free-traction boundary conditions

on the top and bottom surfaces of the plate. By making

a further assumption, the present theory contains only

four unknowns and its governing equations is there-

fore reduced. Equations of motion are derived from

Hamilton’s principle and Navier-type analytical solu-

tions for simply-supported plates are compared with

the existing solutions to verify the validity of the

developed theory. The material properties are contin-

uously varied through the plate thickness by the

power-law and exponential form. Numerical results

are obtained to investigate the effects of the power-law

index and side-to-thickness ratio on the deflections,

stresses, critical buckling load and natural frequencies.

Keywords Functionally graded plates �Bending � Buckling � Vibration

1 Introduction

Functionally graded material (FGM) is an advanced

composite material whose compositions vary accord-

ing to the required performance. It is produced by a

continuously graded variation of the volume fractions

of the constituents (Koizumi 1997), the FGM is thus

suitable for various applications, such as thermal

coatings of barrier for ceramic engines, gas turbines,

nuclear fusions, optical thin layers, biomaterial elec-

tronics, etc.

In recent years, many functionally graded (FG)

plate structures which have been applied for engi-

neering fields led to the development of various plate

theories to accurately predict the bending, buckling

and vibration behaviors of FG plates (Jha et al. 2013).

The classical plate theory (CPT) known as the simplest

one which neglects the transverse shear deformation

effect (Feldman and Aboudi 1997; Javaheri and

Eslami 2002; Mahdavian 2009; Mohammadi et al.

2010; Chen et al. 2006; Baferani et al. 2011) gives only

convenable results for thin FG plates. For FG thick and

moderately thick plates, the first-order shear defor-

mation theory (FSDT) has been used (Praveen and

Reddy 1998; Croce and Venini 2004; Efraim and

Eisenberger 2007; Zhao et al. 2009a, b; Hosseini-

Hashemi et al. 2011; Naderi and Saidi 2010). In a such

approach, in-plane displacements are linearly varied in

the thickness and require a shear correction factor to

correct the unrealistic variation of the transverse shear

stresses and shear strains through the thickness.

T.-K. Nguyen (&)

Faculty of Civil Engineering and Applied Mechanics,

University of Technical Education Ho Chi Minh City, 1

Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City,

Vietnam

e-mail: [email protected]

123

Int J Mech Mater Des

DOI 10.1007/s10999-014-9260-3

Alternatively, higher-order shear deformation theories

(HSDTs) with higher-order variations of displace-

ments have been developed for FG plates (Reddy

2000; Pradyumna and Bandyopadhyay 2008; Jha et al.

2013; Neves et al. 2012, 2012, 2013; Reddy 2011;

Talha and Singh 2010; Zenkour 2006, 2013; Chen

et al. 2009; Mantari and Soares 2012, 2013; Matsu-

naga 2008; Thai 2013), they can predict more accurate

the behaviors of moderate and thick FG plates, and no

shear correction factors are required. However prac-

tically, some of these HSDTs are computational costs

because of number of additional variables introduced

to the theory (Pradyumna and Bandyopadhyay 2008;

Jha et al. 2013; Neves et al. 2012, 2012, 2013; Reddy

2011; Talha and Singh 2010). As a consequence, a

simple higher-order shear deformation theory pro-

posed in this paper is necessary.

This paper aims to develop a simple higher-order

shear deformation theory for bending, free vibration

and buckling analysis of FG plates. By making a

further assumption to the existing higher-order shear

deformation theory, the present theory contains only

four unknowns and its governing equations is there-

fore reduced. Hamilton’s principle is used to derive

equations of motion and Navier-type analytical solu-

tions for simply-supported plates are compared with

the existing solutions to verify the validity of the

developed theory. The material properties are contin-

uously varied through the plate thickness by the

power-law and exponential form. Numerical results

are obtained to investigate the effects of the power-law

index and side-to-thickness ratio on the deflections,

stresses, critical buckling load and natural frequencies.

2 Theoretical formulation

Consider a FG plate as shown in Fig. 1 having the

thickness h, length a and width b, and boundaries with

a suitable regularity. The FG plate is constituted by a

mixture of ceramic and metal components whose

material properties vary through the plate thickness

according to the volume fractions of the constituents.

2.1 Effective material properties of FG plates

The effective elastic material properties of FGMs can

be estimated by continuous model and discrete model.

The first model assumes a continuous material

distribution in the thickness direction without taking

into account the microstructure, whereas the second

one takes into account the microstructure with ideal-

ized geometries. Practically however, in both models,

FG plates are first homogenized with their effective

moduli such as Young’s modulus, Poisson’s ratio,

mass density..., etc, and then their effective properties

are derived from homogeneous plate theories. The

continuous model is most used to study the FGM,

among the approximation of Voigt Hill (1952) is

widely used by its simplification. The works of Reiter

and Dvorak (1997, 1998) showed that the Mori–

Tanaka’s scheme Benveniste (1987) is convenient for

estimating the effective material properties of FGM,

especially in the matrix-inclusion region. In practice,

the Mori–Tanaka’s scheme is more complicated than

Voigt’s model. It should be noted that these estima-

tions based on bounds are efficient if the material

contrast of constituents is not too large. Many more

approximations of the effective elastic material prop-

erties can be found in Gasik (1998). In this paper,

material properties through the thickness are estimated

by two homogenization schemes: power-law form and

exponential form. For power-law form, the effective

material properties of FG plates are expressed by

Reddy (2000):

PðzÞ ¼ ðPc � PmÞVc þ Pm; ð1Þ

where Pc and Pm are the Young’s moduli (E),

Poisson’s ratio (m) and mass density (q) of ceramic

and metal materials located at the top and bottom

surfaces, respectively. The volume fraction of ceramic

material Vc is given as follows:

VcðzÞ ¼2zþ h

2h

� �p

; ð2Þ

where p is the power-law index, which is positive and

z 2 ½� h2; h

2�. Distribution of ceramic material through

Fig. 1 Geometry of a functionally graded plate

T.-K. Nguyen

123

the plate thickness is displayed in Fig. 2. Moreover,

the effective Young’s modulus of FG plates can be

directly calculated according to the exponential law

(Zenkour 2007):

EðzÞ ¼ E0epðz=hþ0:5Þ; ð3Þ

where E0 is Young’s modulus of homogeneous

material. For vibration analysis, the mass density at

location z is varied with respect to the power-law form

Eq. (3).

2.2 Kinematics and strains

The displacement field of the HSDT can be written as:

uðx; y; zÞ ¼ u0ðx; yÞ � z ow0

oxþ f ðzÞhxðx; yÞ

vðx; y; zÞ ¼ v0ðx; yÞ � z ow0

oyþ f ðzÞhyðx; yÞ

wðx; y; zÞ ¼ w0ðx; yÞ;ð4Þ

where u0; v0;w0, hx, hy are five unknown displace-

ments of the midplane of the plate, f ðzÞ represents

shape function defining the distribution of the trans-

verse shear strains and stresses along the thickness. By

assuming that hx ¼ �ouðx; yÞ=ox and hy ¼�ouðx; yÞ=oy (Thai et al. 2014), the displacement

field of the present theory can be rewritten in a simpler

form as:

uðx; y; zÞ ¼ u0ðx; yÞ � z ow0

ox� f ðzÞ ou

ox

vðx; y; zÞ ¼ v0ðx; yÞ � z ow0

oy� f ðzÞ ou

oy

wðx; y; zÞ ¼ w0ðx; yÞ;ð5Þ

where the shape function f ðzÞ is chosen according to

Grover et al. (2013) as:

f ðzÞ ¼ sinh�1 3z

h

� �� z

6

hffiffiffiffiffi13p : ð6Þ

It can be seen that the displacement field in Eq. (5)

contains only four unknowns (u0; v0;w0;u). The strain

field associated with the displacement field in Eq. (5)

are written under following compact form:

� ¼ �0 þ zjb þ f js ð7aÞ

c ¼ gc0; ð7bÞ

where g ¼ �df=dz, �0, jb, js, and c0 are membrane

strains, curvatures and transverse shear strains, respec-

tively. They are related to the displacement field in

Eq. (5) as follows:

�0 ¼ �0xx�

0yyc

0xy

� �¼ ou0

ox

ov0

oy

ou0

oyþ ov0

ox

� �;

jb ¼ jbxxj

byyj

bxy

� �¼ � o2w0

ox2� o2w0

oy2� 2

o2w0

oxoy

� �;

js ¼ jsxxj

syyj

sxy

� �¼ � o2u

ox2� o2u

oy2� 2

o2uoxoy

� �

ð8aÞ

c0 ¼c0

xz

c0yz

( )¼

ouox

ouoy

( )ð8bÞ

The linear constitutive relations of the FG plates are

written as:

rxx

ryy

rxy

8><>:

9>=>; ¼

C11 C12 0

C12 C22 0

0 0 C66

264

375

�xx

�yy

cxy

8><>:

9>=>; ð9aÞ

rxz

ryz

( )¼

C55 0

0 C44

" #cxz

cyz

( )ð9bÞ

where

C11ðzÞ ¼ C22ðzÞ ¼EðzÞ

1� mðzÞ2;C12ðzÞ ¼ mðzÞC11ðzÞ

ð10aÞ

C44ðzÞ ¼ C55ðzÞ ¼ C66ðzÞ ¼EðzÞ

2ð1þ mðzÞÞ ð10bÞ

2.3 Equations of motion

Hamilton’s principle is herein used to derive the

equations of motion:

0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Volume fraction Vc

z/h

p=10

p=5

p=2

p=1

p=0.5

p=0.2

p=0.1

Fig. 2 Volume fraction of ceramic material

Higher-order hyperbolic shear deformation plate model

123

0 ¼ZT

0

ðdU þ dV � dKÞ dt; ð11Þ

where dU, dV , dK are the variations of strain energy,

work done, and kinetic energy of the plate, respec-

tively. The variation of strain energy is calculated by:

dU ¼ZA

Zh=2

�h=2

rxxd�xx þ ryyd�yy þ rxydcxy

þrxzdcxz þ ryzdcyz

dA dz

¼ZA

Nxx

odu0

ox�Mb

xx

o2dw0

ox2�Ms

xx

o2duox2

�

þNyy

odv0

oy�Mb

yy

o2dw0

oy2�Ms

yy

o2duoy2

þ Nxy

odu0

oyþ odv0

ox

� �� 2Mb

xy

o2dw0

oxoy

�2Msxy

o2duoxoy

þ Qx

oduoxþ Qy

oduoy

�dA; ð12Þ

where N, M, and Q are the stress resultants defined by:

ðNxx;Nyy;NxyÞ ¼Zh=2

�h=2

ðrxx; ryy; rxyÞ dz ð13aÞ

ðMbxx;M

byy;M

bxyÞ ¼

Zh=2

�h=2

z ðrxx; ryy; rxyÞ dz ð13bÞ

ðMsxx;M

syy;M

sxyÞ ¼

Zh=2

�h=2

f ðrxx; ryy; rxyÞ dz ð13cÞ

ðQx;QyÞ ¼Zh=2

�h=2

gðrxz; ryzÞ dz ð13dÞ

The variation of work done by in-plane and transverse

loads is given by:

dV ¼ �ZA

�Ndw0dA�ZA

qdw0dA; ð14Þ

where

�N ¼ N0xx

o2w0

ox2þ 2N0

xy

o2w0

oxoyþ N0

yy

o2w0

oy2

The variation of kinetic energy is determined by:

dK ¼ZV

Zh=2

�h=2

ð _ud _uþ _vd _vþ _wd _wÞqðzÞ dA dz

¼ZA

�I0 _u0d _u0 þ _v0d _v0 þ _w0d _w0ð Þ

� I1 _u0

od _w0

oxþ o _w0

oxd _u0 þ _v0

od _w0

oyþ o _w0

oyd _v0

� �

þ I2

o _w0

ox

od _w0

oxþ o _w0

oy

od _w0

oy

� �

� J1 _u0

od _uoxþ o _u

oxd _u0 þ _v0

od _uoyþ o _u

oyd _v0

� �

þ K2

o _uox

od _uoxþ o _u

oy

od _uoy

� �þ J2

o _w0

ox

od _uox

�

þ o _uox

od _w0

oxþ o _w0

oy

od _uoyþ o _u

oy

od _w0

oy

��dA;

ð15Þ

where the dot-superscript convention indicates the

differentiation with respect to the time variable t, qðzÞis the mass density, and the inertia terms Ii, Ji, Ki are

expressed by:

ðI0; I1; I2Þ ¼Zh=2

�h=2

ð1; z; z2ÞqðzÞdz ð16aÞ

ðJ1; J2;K2Þ ¼Zh=2

�h=2

ðf ; zf ; f 2ÞqðzÞdz ð16bÞ

Substituting Eqs. (12), (14), and (15) into Eq. (11),

integrating by parts, and collecting the coefficients of

du0, dv0, dw0; du; the following equations of motion

are obtained:

du0 :oNxx

oxþ oNxy

oy¼ I0 €u0 � I1

o €w0

ox� J1

o €uox

ð17aÞ

dv0 :oNxy

oxþ oNyy

oy¼ I0€v0 � I1

o €w0

oy� J1

o €uoy

ð17bÞ

T.-K. Nguyen

123

dw0 :o2Mb

xx

ox2þ 2

o2Mbxy

oxoyþ

oMbyy

oy2þ �N þ q

¼ I0 €w0 þ I1

o€u0

oxþ o€v0

oy

� �� I2r2 €w0 � J2r2 €u

ð17cÞ

du :o2Ms

xx

ox2þ 2

o2Msxy

oxoyþ

oMsyy

oy2þ oQx

oxþ oQy

oy

¼ J1

o€u0

oxþ o€v0

oy

� �� J2r2 €w0 � K2r2 €u; ð17dÞ

where r2 ¼ o2=ox2 þ o2=oy2 is the Laplacian opera-

tor in two-dimensional Cartesian coordinate system.

Substituting Eq. (7a) into Eq. (9a) and the subsequent

results into Eqs. (13a), (13b) and (13c), the stress

resultants are obtained in terms of strains as following

compact form:

N

Mb

Ms

8><>:

9>=>; ¼

A B Bs

B D Ds

Bs Ds Hs

264

375

�0

jb

js

8><>:

9>=>; ð18Þ

where A;B;D;Bs;Ds;Hs are the stiffnesses of the FG

plate given by:

ðA;B;D;Bs;Ds;HsÞ ¼Zh=2

�h=2

ð1; z; z2; f; zf; f2ÞCðzÞdz

ð19Þ

Similarly, using Eqs. (7b), (9b) and (13d), the

transverse shear forces can be calculated from the

constitutive equations as:

Qx

Qy

� �¼

As55 0

0 As44

� �c0

xz

c0yz

( )ð20Þ

or in a compact form as:

Q ¼ As c0; ð21Þ

where the shear stiffnesses As of the FG plate are

defined by:

As44¼As

55¼Zh=2

�h=2

g2ðzÞC44ðzÞdz¼Zh=2

�h=2

g2ðzÞC55ðzÞdz

ð22Þ

By substituting Eqs. (18) and (21) into Eq. (17a–17d),

the equations of motion can be expressed in terms of

displacements (u0;v0;w0;u) as follows:

A11

o2u0

ox2þ A66

o2u0

oy2þ ðA12 þ A66Þ

o2v0

oxoy

� B11

o3w0

ox3� ðB12 þ 2B66Þ

o3w0

oxoy2� Bs

11

o3uox3

� ðBs12 þ 2Bs

66Þo3u

oxoy2¼ I0 €u0 � I1

o €w0

ox� J1

o €uox

ð23aÞ

A22

o2v0

oy2þ A66

o2v0

ox2þ ðA12 þ A66Þ

o2u0

oxoy� B22

o3w0

oy3

� ðB12 þ 2B66Þo3w0

ox2oy� Bs

22

o3uoy3

� ðBs12 þ 2Bs

66Þo3u

ox2oy¼ I0€v0 � I1

o €w0

oy� J1

o €uoy

ð23bÞ

B11

o3u0

ox3þ ðB12 þ 2B66Þ

o3u0

oxoy2þ ðB12 þ 2B66Þ

o3v0

ox2oy

þ B22

o3v0

oy3� D11

o4w0

ox4� D22

o4w0

oy4� 2ðD12

þ 2D66Þo4w0

ox2oy2� Ds

11

o4uox4� Ds

22

o4uoy4� 2ðDs

12

þ 2Ds66Þ

o4uox2oy2

þ �NðwÞ þ q

¼ I0 €w0 þ I1

o€u0

oxþ o€v0

oy

� �� I2r2 €w0 � J2r2 €u

ð23cÞ

Bs11

o3u0

ox3þ ðBs

12 þ 2Bs66Þ

o3u0

oxoy2þ ðBs

12 þ 2Bs66Þ

o3v0

ox2oy

þ Bs22

o3v0

oy3� Ds

11

o4w0

ox4� Ds

22

o4w0

oy4� 2ðDs

12

þ 2Ds66Þ

o4w0

ox2oy2þ As

55

o2uox2þ As

44

o2uoy2� Hs

11

o4uox4

� 2ðHs12 þ 2Hs

66Þo4u

ox2oy2� Hs

22

o4uoy4

¼ J1

o€u0

oxþ o€v0

oy

� �� J2r2 €w0 � K2r2 €u

ð23dÞ

2.4 Analytical solution for simply-supported FG

plates

The Navier solution procedure is used to obtain the

analytical solutions for which the displacement func-

tions are expressed as product of undetermined

coefficients and known trigonometric functions to


123

satisfy the governing equations and boundary

conditions.

u0ðx; y; tÞ ¼X1m¼1

X1n¼1

u0mn cos kx sin ly eixt ð24aÞ

v0ðx; y; tÞ ¼X1m¼1

X1n¼1

v0mn sin kx cos ly eixt ð24bÞ

w0ðx; y; tÞ ¼X1m¼1

X1n¼1

x0mn sin kx sin ly eixt ð24cÞ

uðx; y; tÞ ¼X1m¼1

X1n¼1

y0mn sin kx sin ly eixt; ð24dÞ

where k ¼ mp=a, l ¼ np=b, xis the frequency of free

vibration of the plate,ffiffiip¼ �1 the imaginary unit.

The transverse load q is also expanded in the double-

Fourier sine series as:

qðx; yÞ ¼X1m¼1

X1n¼1

qmn sin kx sin ly ; ð25Þ

where qmn = q0 for sinusoidally distributed load.

Assuming that the plate is subjected to in-plane

compressive loads of form: N0xx ¼ �N0, N0

yy ¼ �cN0

(here c is non-dimensional load parameter), N0xy ¼ 0.

Substituting Eqs. (24a–24d) and (25) into Eq. (23a–

23d) and collecting the displacements and acceleration

for any values of m and n, the following problem is

obtained:

k11 k12 k13 k14

k12 k22 k23 k24

k13 k23 k33 þ a k34

k14 k24 k34 k44

26664

37775

0BBB@

�x2

m11 0 m13 m14

0 m22 m23 m24

m13 m23 m33 m34

m14 m24 m34 m44

26664

37775

1CCCA

u0mn

v0mn

x0mn

y0mn

8>>><>>>:

9>>>=>>>;

¼

0

0

qmn

0

8>>><>>>:

9>>>=>>>;;

ð26Þ

where

k11 ¼ A11k2 þ A66l2; k12 ¼ ðA12 þ A66Þkl;

k13 ¼ �B11k3 � ðB12 þ 2B66Þkl2

k14 ¼ �Bs11k

3 � ðBs12 þ 2Bs

66Þkl2;

k22 ¼ A66k2 þ A22l2;

k23 ¼ �B22l3 � ðB12 þ 2B66Þk2l

k24 ¼ �Bs22l

3 � ðBs12 þ 2Bs

66Þk2l;

k33 ¼ D11k4 þ 2ðD12 þ 2D66Þk2l2 þ D22l4

k34 ¼ Ds11k

4 þ 2ðDs12 þ 2Ds

66Þk2l2 þ Ds22l

4

k44 ¼ Hs11k

4 þ 2ðHs12 þ 2Hs

66Þk2l2 þ Hs22l

4

þAs55k

2 þ As44l

2

m11 ¼ m22 ¼ I0; m13 ¼ �kI1; m14 ¼ �kJ1;

m23 ¼ �lI1; m24 ¼ �lJ1

m33 ¼ I0 þ I2ðk2 þ l2Þ; m34 ¼ J2ðk2 þ l2Þ;m44 ¼ K2ðk2 þ l2Þa ¼ �N0ðk2 þ cl2Þ

ð27Þ

Eq. (26) is a general form for bending, buckling and

free vibration analysis of FG plates under in-plane and

transverse loads. In order to solve bending problem,

the in-plane compressive load N0 and mass matrix M

are set to zeros. Moreover, the stability problem can be

carried out by neglecting the mass matrix and

transverse load while the free vibration problem is

achieved by omitting the transverse load.

3 Numerical examples

Consider a simply supported FG rectangular plate with

in-plane lengths, a and b in the x� and y� directions,

respectively (Fig. 1). FG plates made of three material

combinations of metal and ceramic: Al/ZrO2, Al/

Al2O3 and Al/SiC are considered. Their material

Table 1 Material properties of metal and ceramic

Material Young’s

modulus

(GPa)

Mass density

(kg/m3)

Poisson’s

ratio

Aluminum (Al) 70 2,702 0.3

Zirconia (ZrO2) 151 3,000 0.3

Alumina (Al2O3) 380 3,800 0.3

Silicon carbide (SiC) 420 3,210 0.3

T.-K. Nguyen

123

properties are given in Table 1. A number of

numerical examples are analyzed in the sequel to

verify the accuracy of present study and investigate

effects of the power-law index and side-to-thickness

ratio on the deflections, stresses, natural frequencies

and critical buckling loads of FG plates. Unless

special mention, the effective material properties are

calculated by the power-law form (Eq. (3)). For

convenience, the following non-dimensional param-

eters are used:

�u¼100Ech3

q0a4u 0;

b

2;z

� �; �w¼10Ech3

q0a4w

a

2;b

2

� �ð28Þ

�rxxðzÞ ¼h

q0arxx

a

2;b

2; z

� �;

�rxyðzÞ ¼h

q0arxz 0; 0; zð Þ;

�rxzðzÞ ¼h

q0arxz 0;

b

2; z

� �ð29Þ

Ncr ¼Ncra

2

D11 � B211=A11

; �Ncr ¼Ncra

2

Emh3ð30Þ

x ¼ xh

ffiffiffiffiffiqc

Ec

r; �x ¼ xa2

h

ffiffiffiffiffiqc

Ec

r;

�b ¼ xab

p2h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi12ð1� m2

cÞqc

Ec

s ð31Þ

Table 2 Comparison of the nondimensional stress and displacements of Al/Al2O3 square plates (a/h = 10)

p Theory �u �w �rxxðh=3Þ �rxyð�h=3Þ �rxzðh=6Þ

1 Quasi-3D Carrera et al. (2008) 0.6436 0.5875 1.5062 0.6081 0.2510

Quasi-3D Wu and Chiu (2011) 0.6436 0.5876 1.5061 0.6112 0.2511

SSDT Zenkour (2006) 0.6626 0.5889 1.4894 0.6110 0.2622

HSDT Mantari et al. (2012) 0.6398 0.5880 1.4888 0.6109 0.2566

TSDT Wu and Li (2010) 0.6414 0.5890 1.4898 0.6111 0.2599

HSDT Thai and Kim (2013) 0.6414 0.5890 1.4898 0.6111 0.2608

Present 0.6401 0.5883 1.4892 0.6110 0.2552



SSDT Zenkour (2006) 0.9281 0.7573 1.3954 0.5441 0.2763


TSDT Wu and Li (2010) 0.8984 0.7573 1.3960 0.5442 0.2721

HSDT Thai and Kim (2013) 0.8984 0.7573 1.3960 0.5442 0.2737

Present 0.8961 0.7567 1.3947 0.5439 0.2721



SSDT Zenkour (2006) 1.0941 0.8819 1.1783 0.5667 0.2580


TSDT Wu and Li (2010) 1.0502 0.8815 1.1794 0.5669 0.2519

HSDT Thai and Kim (2013) 1.0502 0.8815 1.1794 0.5669 0.2537

Present 1.0466 0.8818 1.1766 0.5664 0.2593



SSDT Zenkour (2006) 1.1340 0.9750 0.9466 0.5856 0.2121


TSDT Wu and Li (2010) 1.0763 0.9747 0.9477 0.5858 0.2087

HSDT Thai and Kim (2013) 1.0763 0.9746 0.9477 0.5858 0.2088

Present 1.0719 0.9744 0.9444 0.5852 0.2117


123

3.1 Results of bending analysis

Example 1 The purpose of the first example is to

verify the validity of the present theory in predicting

the bending behaviors. The center deflections, in-

plane and transverse shear stresses of Al/Al2O3 plates

under sinusoidal loads are calculated in Tables 2, 3.

The present results are compared with those predicted

by different shear deformation theories: third-order

shear deformation plate theory (TSDT), sinusoidal

Table 3 Comparison of the nondimensional deflection ( �w) of Al/Al2O3 square plates with material distribution according to the

exponential form

a/h b/a Theory Power-law index

0.1 0.3 0.5 0.7 1 1.5

2 1 3D Zenkour (2007) 0.5769 0.5247 0.4766 0.4324 0.3727 0.2890

Quasi-3D Zenkour (2007) 0.5731 0.5181 0.4679 0.4222 0.3612 0.2771

Quasi-3D Mantari and Soares (2012) 0.5776 0.5222 0.4716 0.4255 0.3640 0.2792

HSDT Mantari et al. (2012) 0.6363 0.5752 0.5195 0.4687 0.4018 0.3079

HSDT Thai and Kim (2013) 0.6362 0.5751 0.5194 0.4687 0.4011 0.3079

Present 0.6211 0.5615 0.5073 0.4579 0.3921 0.3014

2 3D Zenkour (2007) 1.1944 1.0859 0.9864 0.8952 0.7727 0.6017

Quasi-3D Zenkour (2007) 1.1880 1.0740 0.9701 0.8755 0.7494 0.5758



HSDT Thai and Kim (2013) 1.2775 1.1553 1.0441 0.9431 0.8086 0.6238

Present 1.2569 1.1367 1.0275 0.9284 0.7965 0.6153

3 3D Zenkour (2007) 1.4430 1.3116 1.1913 1.0812 0.9334 0.7275

Quasi-3D Zenkour (2007) 1.4354 1.2977 1.1722 1.0580 0.9057 0.6962



HSDT Thai and Kim (2013) 1.5340 1.3873 1.2540 1.1329 0.9719 0.7506

Present 1.5115 1.3671 1.2360 1.1169 0.9587 0.7414

4 1 3D Zenkour (2007) 0.3490 0.3168 0.2875 0.2608 0.2253 0.1805

Quasi-3D Zenkour (2007) 0.3475 0.3142 0.2839 0.2563 0.2196 0.1692



HSDT Thai and Kim (2013) 0.3602 0.3259 0.2949 0.2668 0.2295 0.1785

Present 0.3575 0.3235 0.2927 0.2649 0.2280 0.1775

2 3D Zenkour (2007) 0.8153 0.7395 0.6708 0.6085 0.5257 0.4120

Quasi-3D Zenkour (2007) 0.8120 0.7343 0.6635 0.5992 0.5136 0.3962



HSDT Thai and Kim (2013) 0.8325 0.7534 0.6819 0.6173 0.5319 0.4150

Present 0.8285 0.7498 0.6787 0.6145 0.5296 0.4135

3 3D Zenkour (2007) 1.0134 0.9190 0.8335 0.7561 0.6533 0.5121

Quasi-3D Zenkour (2007) 1.0094 0.9127 0.8248 0.7449 0.6385 0.4927



HSDT Thai and Kim (2013) 1.0325 0.9345 0.8459 0.7659 0.6601 0.5154

Present 1.0281 0.9305 0.8424 0.7628 0.6576 0.5137

T.-K. Nguyen

123

shear deformation plate theory (SSDT), hyperbolic

shear deformable plate theory (HSDT), and quasi-3D

ones which included both transverse shear and normal

deformations. It is observed that the agreements of the

obtained results with those reported by Zenkour (2006)

(SSDT), Thai and Kim (2013) and Mantari et al. (2012)

(HSDT), Wu and Li (2010) (TSDT) are found for both

power-law and exponential form. Moreover in many

cases, the present solutions are better predictions with

quasi-3D ones than TSDT, SSDT and HSDT ones. The

variations of in-plane displacement, in-plane and out-of-

plane stresses through the thickness of Al/Al2O3 square

plate are displayed in Fig. 3. It can be seen that the

maximum axial stress increases with p while it appears

minimum compressive stresses located inside of the

−1.5 −1 −0.5 0 0.5 1 1.5 2−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

u

z/h

p=0p=0.5p=1p=5p=10p=20

u)

−2 0 2 4 6 8−0.5

0

0.5

−0.4

−0.3

−0.2

−0.1

0.1

0.2

0.3

0.4

σxx

z/h

p=0p=0.5p=1p=5p=10p=20

σxx)

−4 −3 −2 −1 0 1 2−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

σxy

z/h

p=0p=0.5p=1p=5p=10p=20

σxy)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

σxz

z/h

p=0p=0.5p=1p=5p=10p=20

(a) In-plane displacement (¯ (b) In-plane stress (¯

(c) In-plane shear stress (¯ (d) Transverse shear stress (σxz)

Fig. 3 Variation of displacement and stresses through the thickness of Al/Al2O3 square plates (a/h = 10)

1020

3040

50

205

1015

200

1

2

3

a/hp

w

Fig. 4 Effect of the side-to-thickness ratio a=h and power-law

index p on the nondimensional center deflection ( �w) of Al/Al2O3

square plates


123

plate for some values of p (p� 1). The maximum shear

stress is located at the mid-plane for homogeneous

plates and tends to lightly move to the upper surface

with respect to p, that is asymmetric characteristic of

FGM through the plate thickness. A 3D interaction

diagram of the power-law index p, side-to-thickness

ratio a=h and center deflection �w is plotted in Fig. 4. It

is noted from this figure that the center deflection

increases with p and decreases with an increase of a=h.

3.2 Results of vibration and buckling analysis

Example 2 This example aims to demonstrate the

accurate of the present theory in predicting the free

0 5 10 15 202

4

6

8

10

12

14

16

18

20

22

p

Non

dim

ensi

onal

nat

ural

freq

uenc

ies

m=1, n=1m=1, n=2m=2, n=2

(a) a/h=10

5 10 15 20 25 30 35 40 45 5022

2.5

3

3.5

4

4.5

5

5.5

6

a/h

Non

dim

ensi

onal

fund

amen

tal f

requ

ency

p=0p=0.5p=1p=5p=10p=20

(b)

Fig. 5 Effect of the power-law index p and side-to-thickness ratio a=h on on the natural frequency ( �x) of Al/Al2O3 square plates

1020

3040

50

205

1015

202

3

4

5

6

a/hp

ω

Fig. 6 Effect of the side-to-thickness ratio a=h and power-law

index p on the nondimensional fundamental frequency ( �x) of

Al/Al2O3 square plates

Table 4 Comparison of the nondimensional fundamental frequency (�b) of Al/ZrO2 square plates

a/h Theory Power-law index

0 0.1 0.2 0.5 1 2 5 10

2 3D Uymaz and Aydogdu (2007) 1.2589 1.2296 1.2049 1.1484 1.0913 1.0344 0.9777 0.9507

Present 1.2571 1.2259 1.2010 1.1443 1.0882 1.0325 0.9771 0.9540


Present 1.7723 1.7241 1.6850 1.6003 1.5245 1.4629 1.4084 1.3726


Present 1.9330 1.8783 1.8342 1.7402 1.6593 1.5994 1.5500 1.5095


Present 1.9824 1.9257 1.8799 1.7830 1.7006 1.6417 1.5945 1.5524


Present 1.9971 1.9398 1.8935 1.7957 1.7129 1.6544 1.6079 1.5653


Present 1.9993 1.9418 1.8955 1.7975 1.7147 1.6562 1.6098 1.5671

T.-K. Nguyen

123

vibration behavior of Al/Al2O3 and Al/ZrO2 plates.

Table 4 presents the comparison of the fundamental

frequency of Al/ZrO2 square plates derived from the

present study and 3D plate model Uymaz and

Aydogdu (2007). It can be seen that the obtained

results agree very well with 3D solution. Effects of the

power-law index, side-to-thickness ratio and aspect

ratio are summarized in Tables 5 and 6. They are

compared with solutions of FSDT Hosseini-Hashemi

et al. (2011), TSDT Hosseini-Hashemi et al. (2011),

HSDT Thai and Kim (2013) and quasi-3D Matsunaga

(2008). It is observed that the present results are again

found more close in many cases to 3D-quasi plate

model than SSDT, TSDT and HSDT. The variation of

natural frequencies in terms of the power-law index

and side-to-thickness ratio is plotted in Fig. 5. It can be

seen from this figure that the natural frequencies

decrease with the increase of the power-law index. It is

due to the fact that a higher value of p corresponds to

lower value of volume fraction of the ceramic phase,

and thus makes the plates become the softer ones.

Figure 5b shows that with an increase of the side-to-

thickness ratio, the shear deformation effect becomes

very effective in a relatively large region (b=h� 30).

A 3D interaction diagram of the power-law index,

side-to-thickness ratio and fundamental frequency is

also presented in Fig. 6.

Example 3 The next example investigates buckling

responses of Al/Al2O3 and SiC plates, three types of in-

plane loads are considered: uniaxial compression

(c=0), biaxial compressions (c=1) and axial compres-

sion and tension (c=-1). It should be noted that the

stretching-bending coupling exists in FG plates due to

the variation of material properties through the

thickness. This coupling produces deflection and

bending moments when the plate is subjected to in-

plane compressive loads. Hence, the bifurcation-type

buckling will not occur Liew et al. (2003); Qatu and

Leissa (1993). However, for movable-edge plate, the

bifurcation-type buckling occurs when the in-plane

loads are applied at the neutral surface (Naderi and

Saidi 2010; Aydogdu 2008). Therefore, the buckling

analysis is presented herein for the FG plate subjected

to in-plane loads acting on the neutral surface (Thai

and Vo 2013). The obtained results are given in Tables

7 and 8. It is clear that the results of present study again

agree well with previous solutions FSDT Mohammadi

et al. (2010), HSDT Bodaghi and Saidi (2010) and

HSDT Thai and Choi (2012). Figure 7 shows the

critical buckling loads of rectangular plates with

respect to the power-law index. It is observed from

this figure that they decrease with the increase of the

power-law index, and increase with the side-to-

thickness ratio up to the point b=h ¼ 30 from which

the curves become flatter.

Example 4 The last example presents the lowest

load-frequency curves (Fig. 8) for both homogeneous

and FG rectangular plates (a=b ¼ 0:5). It can be seen

that all fundamental frequencies diminish as in-plane

0 5 10 15 201

2

3

4

5

6

7

8

9

p

Non

dim

ensi

onal

crit

ical

buc

klin

g lo

ad

γ=0γ=1γ=−1

(a) a/h=5

5 10 15 20 25 30 35 40 45 501

2

3

4

5

6

7

a/h

Non

dim

ensi

onal

crit

ical

buc

klin

g lo

ad

p=0

p=5

p=1

p=0.5

p=20

p=10

(b) γ = 1

Fig. 7 Effect of the power-law index p and side-to-thickness ratio a=h on the critical buckling load ( �Ncr) of Al/Al2O3 rectangular plates

(a/b = 0.5)


123

loads change from tension to compression. In com-

pression region ( �Ncr [ 0), the fundamental frequen-

cies are the largest for the plates under uniaxial

compression and tension (c ¼ �1) and the smallest for

ones under biaxial compressive load (c ¼ 1).

However, this order is changed in tension region. It

is from load-frequency curves that the critical buck-

ling loads can be determined indirectly by vibration

analysis through load-frequency curves, which corre-

sponds to zero natural frequencies.

Table 5 Comparison of the first three nondimensional frequencies (x) of Al/Al2O3 square plates

a/h Mode (m,n) Theory Power-law index

0 0.5 1 4 10

5 1(1,1) Quasi-3D Matsunaga (2008) 0.2121 0.1819 0.1640 0.1383 0.1306

TSDT Hosseini-Hashemi et al. (2011) 0.2113 0.1807 0.1631 0.1378 0.1301

FSDT Hosseini-Hashemi et al. (2011) 0.2112 0.1805 0.1631 0.1397 0.1324

HSDT Thai and Kim (2013) 0.2113 0.1807 0.1631 0.1378 0.1301

Present 0.2117 0.1810 0.1634 0.1378 0.1303

2(1,2) Quasi-3D Matsunaga (2008) 0.4658 0.4040 0.3644 0.3000 0.2790



HSDT Thai and Kim (2013) 0.4623 0.3989 0.3607 0.2980 0.2771

Present 0.4645 0.4004 0.3622 0.2981 0.2783

3(2,2) TSDT Hosseini-Hashemi et al. (2011) 0.6688 0.5803 0.5254 0.4284 0.3948


HSDT Thai and Kim (2013) 0.6688 0.5803 0.5254 0.4284 0.3948

Present 0.6734 0.5836 0.5286 0.4291 0.3974

10 1(1,1) Quasi-3D Matsunaga (2008) 0.0578 0.0492 0.0443 0.0381 0.0364



HSDT Thai and Kim (2013) 0.0577 0.0490 0.0442 0.0381 0.0364

Present 0.0577 0.0490 0.0442 0.0381 0.0364

2(1,2) Quasi-3D Matsunaga (2008) 0.1381 0.1180 0.1063 0.0905 0.0859



HSDT Thai and Kim (2013) 0.1377 0.1174 0.1059 0.0903 0.0856

Present 0.1379 0.1175 0.1060 0.0902 0.0857

3(2,2) TSDT Hosseini-Hashemi et al. (2011) 0.2113 0.1807 0.1631 0.1378 0.1301


HSDT Thai and Kim (2013) 0.2113 0.1807 0.1631 0.1378 0.1301

Present 0.2117 0.1810 0.1634 0.1378 0.1303

20 1(1,1) TSDT Hosseini-Hashemi et al. (2011) 0.0148 0.0125 0.0113 0.0098 0.0094


HSDT Thai and Kim (2013) 0.0148 0.0125 0.0113 0.0098 0.0094

Present 0.0148 0.0125 0.0113 0.0098 0.0094

2(1,2) Present 0.0365 0.0310 0.0279 0.0241 0.0231

3(2,2) Present 0.0577 0.0490 0.0442 0.0381 0.0364

T.-K. Nguyen

123

4 Conclusions

A higher-order hyperbolic shear deformation plate

model for analysis of functionally graded plates has

been proposed in this paper. The theory accounts for

hyperbolic distribution of transverse shear stress, and

satisfies the traction-free boundary conditions on the

top and bottom surfaces of the plate without using

shear correction factor. The proposed theory contains

only four unknowns and equations of motion are

Table 6 Nondimensional natural frequencies ( �x) of Al/Al2O3 square plates

a/b a/h Mode (m,n) Theory Power-law index

0 0.5 1 2 5 10

0.5 5 1 (1,1) Present 3.4464 2.9380 2.6509 2.3971 2.2260 2.1432

2 (1,2) Present 5.2932 4.5258 4.0860 3.6859 3.3919 3.2574

3 (2,2) Present 11.6113 10.0109 9.0538 8.1181 7.2951 6.9568

10 1 (1,1) Present 3.6533 3.0996 2.7946 2.5371 2.3911 2.3118

2 (1,2) Present 5.7731 4.9031 4.4216 4.0105 3.7671 3.6388

3 (2,2) Present 13.7855 11.7519 10.6036 9.5884 8.9042 8.5729

20 1 (1,1) Present 3.7127 3.1455 2.8355 2.5773 2.4401 2.3622

2 (1,2) Present 5.9209 5.0176 4.5234 4.1104 3.8880 3.7629

3 (2,2) Present 14.6131 12.3983 11.1785 10.1482 9.5645 9.2471

1 5 1 (1,1) Present 5.2932 4.5258 4.0860 3.6859 3.3919 3.2574

2 (1,2) Present 11.6113 10.0109 9.0538 8.1181 7.2951 6.9568

3 (2,2) Present 16.8351 14.5888 13.2140 11.8101 10.4647 9.9360

10 1 (1,1) Present 5.7731 4.9031 4.4216 4.0105 3.7671 3.6388

2 (1,2) Present 13.7855 11.7519 10.6036 9.5884 8.9042 8.5729

3 (2,2) Present 21.1728 18.1033 16.3438 14.7435 13.5677 13.0296

20 1 (1,1) Present 5.9209 5.0176 4.5234 4.1104 3.8880 3.7629

2 (1,2) Present 14.6131 12.3983 11.1785 10.1482 9.5645 9.2471

3 (2,2) Present 23.0925 19.6126 17.6865 16.0419 15.0685 14.5550

−2 0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Ncr

ω

γ=0γ=1γ=−1

−0.5 0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

Ncr

ω

γ=0γ=1γ=−1

(a) p=0 (b) p=10

Fig. 8 Effect of in-plane loads on the nondimensional fundamental frequency of Al/Al2O3 rectangular plates (a/b = 0.5, a/h = 10)


123

Table 7 Comparison of the critical buckling load (Ncr) of Al/SiC square plates (a/h = 10)

c Theory Power-law index p

0 0.5 1 2 5 10

0 Present 37.4215 37.6650 37.7560 37.6327 36.8862 36.5934

FSDT Mohammadi et al. (2010) 37.3708 – 37.7132 37.7089 – –

HSDT Bodaghi and Saidi (2010) 37.3714 – 37.7172 37.5765 – –

HSDT Thai and Choi (2012) 37.3721 – 37.7143 37.6042 – –

1 Present 18.7107 18.8325 18.8780 18.8163 18.4431 18.2967




-1 Present 72.3281 73.4526 73.8426 73.2827 69.9876 68.7244




Table 8 Comparison of the critical buckling load ( �Ncr) of Al/Al2O3 plates

c a/b a/h Theory Power-law index p

0 0.5 1 2 5 10

0 0.5 5 HSDT Thai and Choi (2012) 6.7203 4.4235 3.4164 2.6451 2.1484 1.9213

Present 6.7417 4.4343 3.4257 2.6503 2.1459 1.9260

10 HSDT Thai and Choi (2012) 7.4053 4.8206 3.7111 2.8897 2.4165 2.1896

Present 7.4115 4.8225 3.7137 2.8911 2.4155 2.1911


Present 7.6009 4.9307 3.7937 2.9585 2.4942 2.2695

1 5 HSDT Thai and Choi (2012) 16.0211 10.6254 8.2245 6.3432 5.0531 4.4807

Present 16.1003 10.6670 8.2597 6.3631 5.0459 4.4981


Present 18.6030 12.1317 9.3496 7.2687 6.0316 5.4587


Present 19.3593 12.5652 9.6702 7.5386 6.3437 5.7689

1 0.5 5 HSDT Thai and Choi (2012) 5.3762 3.5388 2.7331 2.1161 1.7187 1.5370

Present 5.3934 3.5475 2.7406 2.1202 1.7167 1.5408


Present 5.9292 3.8580 2.9710 2.3129 1.9324 1.7529


Present 6.0807 3.9445 3.0350 2.3668 1.9953 1.8156


Present 8.0501 5.3335 4.1299 3.1815 2.5230 2.2491


Present 9.3015 6.0659 4.6748 3.6344 3.0158 2.7293


Present 9.6796 6.2826 4.8351 3.7693 3.1718 2.8844

T.-K. Nguyen

123

derived from Hamilton’s principle. Navier-type solu-

tions are obtained for simply-supported boundary

conditions and compared with the existing solutions to

verify the validity of the developed theory. The

material properties are estimated by power-law and

exponential form. The effects of the power-law index

and side-to-thickness on the deflection, stresses,

critical buckling load and natural frequencies as well

as load-frequency curves are analyzed. The obtained

results are in well agreement with different higher-

order shear deformation theories and closer to quasi-

3D plate models in many cases. The proposed theory is

found to be appropriate, simple and efficient in

analyzing bending, vibration and buckling problem

of FG plates.

Acknowledgments This research is funded by Vietnam

National Foundation for Science and Technology Development

(NAFOSTED) under Grant No. 107.02-2012.07.

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Table 8 continued

c a/b a/h Theory Power-law index p

0 0.5 1 2 5 10

-1 0.5 5 HSDT Thai and Choi (2012) 8.9604 5.8980 4.5551 3.5268 2.8646 2.5617

Present 8.9890 5.9124 4.5676 3.5337 2.8612 2.5679


Present 9.8820 6.4299 4.9516 3.8548 3.2206 2.9214


Present 10.1345 6.5742 5.0583 3.9447 3.3255 3.0260


Presenta 26.4999 17.9424 13.9872 10.6421 7.9571 6.9626


Presenta 35.9559 23.6497 18.2704 14.1349 11.4447 10.2717


Presenta 39.5280 25.7197 19.8065 15.4190 12.8824 11.6857

a Critical buckling occurs at (m; n) = (2,1)


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Mindlin functionally graded rectangular plates. Int.

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