7. a cell-based smoothed discrete shear gap method (cs-dsg3) based on the c0 type higher order shear...

7/29/2019 7. a Cell-based Smoothed Discrete Shear Gap Method (CS-DSG3) Based on the C0 Type Higher Order Shear Defor

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A cell-based smoothed discrete shear gap method (CS-DSG3) based

on the C0-type higher-order shear deformation theory for static and free

vibration analyses of functionally graded plates

P. Phung-Van a, T. Nguyen-Thoi a,b,, Loc V. Tran a, H. Nguyen-Xuan a,b

a Division of Computational Mechanics, Ton Duc Thang University, Nguyen Huu Tho St., Tan Phong Ward, Dist.7, Hochiminh City, Viet Namb Department of Mechanics, Facultyof Mathematics & ComputerScience,Universityof Science, Vietnam NationalUniversity HCMC, 227NguyenVan Cu,Dist. 5, HochiminhCity, VietNam

a r t i c l e i n f o

Article history:

Received 27 March 2013

Received in revised form 19 May 2013

Accepted 5 June 2013

Available online xxxx

Keywords:

Smoothed finite element methods

ReissnerMindlin plate

Cell-based smoothed discrete shear gap

technique (CS-DSG3)

Functionally graded plates (FGPs)

Thermo-mechanical loads

The higher-order shear deformation plate

theory (HSDT)

a b s t r a c t

A cell-based smoothed discrete shear gap method (CS-DSG3) based on the first-order shear deformation

theory was recently proposed for static and dynamics analyses of Mindlin plates. In this paper, the CS-

DSG3 is extended to the C0-type high-order shear deformation plate theory for the static and free vibra-

tion analyses of functionally graded plates (FGPs). In the FGPs, the material properties are assumed to

vary through the thickness by a simple power rule of the volume fractions of the constituents. In the sta-

tic analysis, both thermal and mechanical loads are considered and a two-step procedure is performed

includinga step of analyzing the temperature field along the thickness of the plate anda step of analyzing

the behavior of the plate subjected to both thermal and mechanical loads. The accuracy and reliability of

the proposed method is verified by comparing its numerical solutions with those of other available

numerical results.

2013 Elsevier B.V. All rights reserved.

1. Introduction

Being first proposed in Sendai (Japan) in 1984, functionally

graded materials (FGM) are then developed rapidly around the

world [14]. The FGMs are classified as special composites whose

material properties change continuously and smoothly along cer-

tain dimensions of the structure according to a predetermined for-

mula. This makes FGMs have many advantages in engineering

applications because the material properties of FGMs can be al-

tered according to the specific requirements. For the functionally

graded plates (FGPs), the volume fractions are derived from a func-

tion of position through their thickness. The FGPs usually consist of

metal and ceramic which make them both tough and resistant to

high temperatures. This property makes the FGPs become more

suitable to apply in aerospace structures, nuclear plants and

semi-conductor technologies.

With the advantageous features of FGPs in many practical appli-

cations and the limitations of analytical methods [59], many

effective numerical methods have been devised to analyse and

simulate the behavior of FGPs. For static analysis, Praveen [10]

investigated the static thermo-elastic response of FGPs using the fi-

nite element method. Lee [11] presented thermoelastic analysis of

FGPs using element-free kp-Ritz. Liew et al. [1215] studied the

thermal behavior of FGPs with temperature-dependent properties.

Reddy [16,17] presented finite element models for FGPs based on

the third-order shear deformation theory. Javaheri et al. [18] de-

rived equilibrium and stability equations of FGPs under thermal

loads based on a high-order shear deformation plate theory

(HSDT). Shen [19] studied the non-linear bending response of FGPs

subjected to thermalmechanical loads, and Woo and Merguid [20]

carried out the non-linear analysis of FGM plates and shells.

For dynamic and buckling analyses, Huang and Shen [21] pre-

sented the non-linear vibration and dynamic response of FGPs in

thermal environments. Yang and Shen [22] studied the free vibra-

tion and dynamic response of FGPs subjected to impulsive lateral

loads combined with initial in-plane actions in a thermal environ-

ment. Vel and Batra [23] provided a three-dimensional solution for

the free and forced vibration of simply supported FGPs by using

different plate theories. Efraim and Eisenberger [24] analysed the

frequency characteristics of thick annular FGPs of variable thick-

ness. Matsunaga [25,26] used a two-dimensional global higher-or-

der deformation theory to analyse free vibration and buckling of

FGPs. Najafizadeh [27] carried out a thermal buckling analysis of

0927-0256/$ - see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.commatsci.2013.06.010

Corresponding author at: Department of Mechanics, Faculty of Mathematics &

Computer Science, University of Science, Vietnam National University HCMC, 227

Nguyen Van Cu, Dist. 5, Hochiminh City, Viet Nam. Tel.: +84 942340411.

E-mail addresses: [email protected], [email protected](T.Nguyen-Thoi).

Computational Materials Science xxx (2013) xxxxxx

Contents lists available at SciVerse ScienceDirect

Computational Materials Science

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m m a t s c i

Please cite this article in press as: P. Phung-Van et al., Comput. Mater. Sci. (2013), http://dx.doi.org/10.1016/j.commatsci.2013.06.010
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circular FGPs based on a HSDT. In addition, some finite element

methods [2830] or meshfree approaches [3135] were also de-

vised to solve the FGM structures. So far, based on the update of

authors, there have not yet been the papers presenting numerical

examples using three-node triangular plate elements based on

the C0-type HSDT [36] for analysis of functionally graded plates

(FGPs) subjected to thermo-mechanical loads. This paper hence

will try to fill this gap by using a new three-node triangular plate

element proposed recently.

In the other front of the development of numerical methods, Liu

and Nguyen-Thoi et al. [37] have integrated the strain smoothing

technique into the FEM to create a series of smoothed FEM (S-

FEM) [38] such as a cell/element-based smoothed FEM (CS-FEM)

[39], a node-based smoothed FEM [40], an edge-based smoothed

FEM [41] and a face-based smoothed FEM [42]. Each of these S-

FEM has different properties and has been used to produce desired

solutions for a wide class of benchmark and practical mechanics

problems. Several theoretical aspects of the S-FEM models have

been provided in Refs [43,44]. The S-FEM models have also been

further investigated and applied to various problems such as plates

and shells [4547], piezoelectricity [48], fluidsolid interaction

[49], visco-elastoplasticity [50,51], limit and shakedown analysis

for solids [52], etc.

Among these S-FEM models, the CS-FEM [38,39] shows some

interesting properties in the solid mechanics problems. Extendingthe idea of the CS-FEM to plate structures, Nguyen-Thoi et al. [54]

have recently formulated a cell-based smoothed stabilized discrete

shear gap element (CSDSG3) for static, and free vibration analyses

of isotropic Mindlin plates by incorporating the CS-FEM with the

original DSG3 element [53]. In the CS-DSG3, each 3-node triangular

element will be divided into three sub-triangles, and in each sub-

triangle, the stabilized DSG3 is used to compute the strains. Then

the strain smoothing technique on whole the triangular element

is used to smooth the strains on these three sub-triangles. The

numerical results showed that the CSDSG3 is free of shear locking

and shows four superior properties such as: (1) be a strong

competitor to many existing three-node triangular plate elements

in the static analysis; (2) can give high accurate solutions for

problems with skew geometries in the static analysis; (3) can givehigh accurate solutions in free vibration analysis; (4) can provide

accurately the values of high frequencies of plates by using only

coarse meshes.

Due to these advantages, in this paper, the CS-DSG3 is further

extended to the C0-HSDT [36] for the static and free vibration anal-

yses of FGPs. In the FGPs, the material properties are assumed to

vary through the thickness by a simple power rule of the volume

fractions of the constituents. In the static analysis, both thermal

and mechanical loads are considered and a two-step procedure is

performed including a step of analyzing the temperature field

along the thickness of the plate and a step of analyzing the behav-

ior of the plate subjected to both thermal and mechanical loads.

The accuracy and reliability of the proposed method is verified

by comparing its numerical solutions with those of other availablenumerical results.

2. Functionally graded plates and thermal distribution

A functionally graded plate made of ceramic and metal is shown

in Fig. 1. The material property is assumed to be graded through

the thickness by the power law distribution expressed as

Pz Pc PmVc Pm 1

Vc 1

2

z

t

nnP 0 2

where subscripts m and c refer to the metal and ceramic constitu-ents, respectively; P represents the effective material properties,

including the Youngs modulus E, density q, Poissons ratio m, ther-mal conductivity k and thermal expansion a; Pc and Pm denote theproperties of the ceramic and metal, respectively; Vc is the volume

fraction of the ceramic; z is the thickness coordinate of plate and

varies from t/2 to t/2; n is the volume fraction exponent. The vol-

ume fraction through the thickness for different volume fraction

exponents n is illustrated in Fig. 2.

In the thermal analysis of the FGPs, it is assumed that the tem-

perature distributions of the ceramic (top) surface and metal (bot-

tom) surface are the constant temperature, and the temperature

varies only through the thickness direction. The temperature vari-

ation along the thickness is attained by solving the one-dimen-

sional steady state heat conduction equation, which is given by

d

dzkz

dT

dz

0 3

with boundary conditions

T Ttop at z t=2

T Tbot at z t=24

Fig. 1. Functionally graded plate.

Fig. 2. Volume fraction Vc versus the thickness.

2 P. Phung-Van et al. / Computational Materials Science xxx (2013) xxxxxx

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where Ttop and Tbot are the top and bottom surface temperatures,

respectively; and k(z) represents the thermal conductivity coeffi-cient at z and is also defined similarly as Eq. (1).

Fig. 3 illustrates the temperature distributions through the

thickness of a FGP made by the Aluminum-Ziconia (Al/ZnO2-1)

for the various value of n, where the top and bottom surfaces are

hold at 300 C and 20 C, respectively. It is evident that tempera-

ture in the FGPs constituted by both ceramic and metal

components.

3. C0type higher-order shear deformation theory and

weakform for FGM Mindlin plates

3.1. C0type HSDT for FGM Mindlin plates

According to C0-type HSDT model [36], the displacements of an

arbitrary point in the plate are expressed by

ux;y;z u0 z4z3

3t2

bx

4z3

3t2/x

vx;y;z v0 z4z3

3t2

by

4z3

3t2/y h=2 6 z6 h=2

wx;y w

5

where tis thickness of plate, u0 = {u0 v0}T and w0 are the membrane

displacements and the transverse displacement of the mid-plane;

and b = {bx by}T are the rotations around y-axis and x-axis, respec-

tively, with the positive directions defined as shown in Fig. 4. Eq.

(5) is developed from Reddys higher-order theory [55] in which,derivative of deflection is replaced by warping function / = {/x/y}

T. Thus, the generalized displacement vector with 5 degrees of

freedom for C1 continuity element can be transformed into the vec-

tor with 7 degrees of freedom for C0 continuity element as

u u0 v0 w0 bx by /x /y T

.

For a FGM Mindlin plate, the strains in plain is expressed by the

following equation

exx eyy cxyT

e0 zj1 z3j2 6

where the membrane strains are obtained from the symmetric dis-

placement gradient

e0

@u0@x

@v0@y

@u0@y

@v0@x

8>>>:9>>=>>; rs u0 7

and the bending strains are given by

j1 12 frb rbTg

j2 k6 fr/ r/T rb rbTg

with k 4

t28

and the transverse shear strains are basically defined as

cxz cyzT es z

2js 9

with

es rw b

js cb /10

where r = [@/@x @/@y]T is the gradient operator.Under the temperature condition, the thermal strain of plate is

expressed as

eth azDTz 1 1 0 11

where DT is the temperature change from a stress-free state.

From Hooks law, the stress in plane is given by

r Ee0 zj1 z3j2 eth 12

and the shear stress is expressed as

s Ges z2js 13

where the material matrices are given by

E Ez

1 mz2

1 mz 0

mz 1 0

0 0 1 mz=2

264375 14

G Ez

21 mz

1 0

0 1

!15

in which E(z) is Youngs modulus; m(z) is the Poissons ratio varyingaccording to the power law as in Eq. (1).

Note that the transverse shear strains in Eq. (9) and the shearstress in Eq. (13) are represented parabolically, hence the shear

correction factor in the C0-type HSDT formulation [36] is not nec-

essary as shown in Eq. (15).

Fig. 3. Temperature distribution through the thickness of the Al/ZnO2-1 FGM plate.

Fig. 4. Positive directions of displacement u, v, w and two rotations bx, by in Mindlin plate.

P. Phung-Van et al. / Computational Materials Science xxx (2013) xxxxxx 3

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3.2. Weakform equation for FGM Mindlin plates

The standard Galerkin weakform of the static equilibrium equa-

tions [56] for the FGM Mindlin plate subjected to thermo-mechan-

ical loads can be written as

ZXdeTp D

epdX

ZXdcTDScdX

ZXdwpdX

ZXdeTrthdX 16

in which p is the transverse loading per unit area; ep, c and rth areexpressed by

ep e0 j1 j2f gT; c es jsf g

T; rth Nth Mth Pth

T

17

and material constant matrices D and DS have the forms of

D

A B E

B D F

E F H

264375; DS AS BS

BS

DS

" #; 18

where

Aij; Bij; Dij; Eij; Fij; Hij Zh=2

h=2

1;z;z2;z3;z4;z6Qijdz i;j 1; 2; 6

Asij; B

sij; D

sij

Zh=2h=2

1;z2;z4Qijdz i;j 4; 5

19

and Nth, Mth, Pth are thermal force and moment resultants and com-

puted by

Nth 1 1 0 T

Zt=2t=2

Ez

1 mzkzDTdz

Mth 1 1 0 T

Zt=2t=2

Ez

1 mzkzzDTdz

Pth 1 1 0 T

Zt=2

t=2

Ez

1 mzkzz3DTdz

20

4. FEM formulation the FGM ReissnerMindlin plate

Now, discretize the bounded domain X into Ne finite elements

such that X SNe

e1Xe andXi \Xj = ;, ij, then the finite element

solution uh u0 v0 w0 bx by /x /y T

of a displacement

model for the FGM Mindlin plate is expressed as

uh

XNn

i1

Nix 0 0 0 0 0 0

0 Nix 0 0 0 0 0

0 0 Nix 0 0 0 0

0 0 0 Nix 0 0 0

0 0 0 0 Nix 0 0

0 0 0 0 0 Nix 0

0 0 0 0 0 0 Nix

2666666666664

3777777777775

di Nd

21

where Nn is the total number of nodes of problem domain discret-

ized; Ni(x) is shape function at the ith node;

di ui vi wi bxi byi /xi /yi T

is the displacement vector

of the nodal degrees of freedom of uh associated to the ith node,

respectively.

Substituting Eq. (21) into Eqs. (7), (8), (10), then the strains in

Eq. (17) can be expressed in the general forms of nodal displace-

ments as

eh ep cT X

Nn

i1

B

id

i22

where eh is the compatible strain and Bi is the generalized straindisplacement matrix expressed by

Bi B

mi

TB

b1i

TB

b2i

TB

s0i

TB

s1i

T !23

in which

Bmi

Ni;x 0 0 0 0 0 0

0 Ni;y 0 0 0 0 0

Ni;y Ni;x 0 0 0 0 0

264 375;B

b1

i

0 0 0 Ni;x 0 0 0

0 0 0 0 Ni;y 0 0

0 0 0 Ni;y Ni;x 0 0

264375;

Bs0i

0 0 Ni;x Ni 0 0 0

0 0 Ni;y 0 Ni 0 0

!;

Bs1i c

0 0 0 Ni 0 Ni 0

0 0 0 0 Ni 0 Ni

!;

24

Bb2

i

c

3

0 0 0 Ni;x 0 Ni;x 0

0 0 0 0 Ni;y 0 Ni;y0 0 0 Ni;y Ni;x Ni;y Ni;x

264 375where Ni,x and Ni,y are the derivatives of the shape functions in x-

direction and y-direction, respectively.

The discretized system of equations of FGM Mindlin plate using

the FEM for static analysis then can be expressed as

Kd F 25

where F is the load vector computed by

F

ZX

pN BmI T

Nth Bb1I

TMth B

b2I

TPth

!dX f

b26

in which fb is the remaining part of F subjected to prescribed

boundary loads; and K is the global stiffness matrix assembled from

the element stiffness matrices Ke as

K XNee1

Ke XNee1

ZXe

BTi D

BjdX

ZXe

STi D

s SjdX

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Ke

27

where Bi and Si are defined by

Bi Bmi

TB

b1

i

TB

b2

i

T !28

Si Bs0i

TB

s1i

Th i29

For free vibration analysis, we have

K x2Md 0 30

where x is the natural frequency and M is the global mass matrixcomputed by

M

ZX

NT

mNTdX 31

in which m is the matrix defined as

m

I1 0 0 I2 0c3

I4 0

I1 0 0 I2 0c3

I4

I1 0 0 0 0

I3 0c3

I5 0

I3 0c3

I5

c3

2I7 0

sym c3 2I7

26666666666664

37777777777775

32




5/16

with

I1; I2; I3; I4; I5; I7

Zh=2h=2

qz1;z;z2;z3;z4;z6dz 33

5. Formulation of a cell-based smoothed discreate shear gap

(CS-DSG3) method based on the C0-type HSDT for FGM Mindlin

plates

5.1. Formulation of the DSG3 based on the C0 -type HSDT

In the original DSG3 [53], the problem domain is discretized

into Nn nodes and Ne three-node triangular elements. The first-or-

der shear deformation plate theory (FSDT) is used for Mindlin

plates behavior and each node only has 3 degrees of freedom

di wi bxi byi T

. The formulation of the DSG3 is based on the

concept shear gap of displacements along the edges of the ele-

ments. In the DSG3, the shear strain is linear interpolated from

the shear gaps of displacement by using the standard element

shape functions. The DSG3 element is shear-locking-free and has

several superior properties as presented in Ref [53]. In the

present paper, the DSG3 is extended to the C0-type HSDT formula-

tion [36] and each node will have 7 degrees of freedom

di ui vi wi bxi byi /xi /yi T

as mentioned in Section 4.

Using a mesh of 3-node triangular elements, the approximation

ue ue ve we bex be

y /e

x /e

y

Tfor a 3-node triangular

element Xe for Mindlin FGM plates can be written as

uex

X3i1

Nei x 0 0 0 0 0 0

0 Nei x 0 0 0 0 0

0 0 Nei x 0 0 0 0

0 0 0 Nei x 0 0 0

0 0 0 0 Nei x 0 0

0 0 0 0 0 Nei x 0

0 0 0 0 0 0 Nei x

2666666666664

3777777777775

|fflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflffl fflffl{zfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflffl}Nei x

dei

X3i1

Nei xd

ei

34

where de

i uei v

ei w

ei b

exi b

eyi /

exi /

eyi

Tare the nodal degrees

of freedom of ue associated to ith node and Nei x

diag Nei x; Nei x; N

ei x; N

ei x; N

ei x; N

ei x; N

ei x

, i = 1, 2, 3, in

which Nei x are the linear shape functions of the 3-node triangular

elementXe.

Substituting Eq. (34) into Eqs. (23) and (24), we obtained the

membrane strain ee0, the curvatures of the deflection je1; j

e2 and

the shear strains ch [53] of the element in the forms of

ee0 Bm

de

35

je1 Bb1 d

e; je2 B

b2 de

36

ce ee

s

jes& ' Bs0

Bs1

& 'de 37where d

e d

e1 d

e2 d

e3

Tis the nodal displacement vector of

element; andBm, Bb1 ; Bb2 ; Bs0 , Bs1 arethe gradient matricesexpressed

by

Bm

1

2Ae

b c 0 0 0 0 0 0

0 d a 0 0 0 0 0

d a b c 0 0 0 0 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

m1

c 0 0 0 0 0 0

0 d 0 0 0 0 0

d c 0 0 0 0 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

m2

b 0 0 0 0 0 0

0 a 0 0 0 0 0

a b 0 0 0 0 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

m3

266664377775 12Ae Bm1 Bm2 Bm3 38

Bb1

1

2Ae

0 0 0 b c 0 0 00 0 0 0 d a 0 0

0 0 0 d a b c 0 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflfflffl}B

b11

0 0 0 c 0 0 00 0 0 0 d00

0 0 0 d c00|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflfflffl}B

b12

0 0 0 b 0 0 00 0 0 0 a00

0 0 0 a b00|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflffl}B

b13

266664377775 12Ae Bb11 Bb12 Bb13

h i39

Bb2

1

2Ae

0 0 0 b c 0 b c 0

0 0 0 0 d a 0 d a

0 0 0 d a b c d a b c|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

b21

0 0 0 c 0 c 0

0 0 0 0 d 0 d

0 0 0 d c d c|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflffl}B

b22

0 0 0 b 0 b 0

0 0 0 0 a 0 a

0 0 0 a b a b|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

b23

266664377775 12Ae Bb21 Bb22 Bb23

h i

40

Bs0 12Ae

0 0 b c Ae 0 0 0

0 0 d a 0 Ae 0 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

s01

0 0 c ac=2 bc=2 0 0

0 0 d ad=2 bd=2 0 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflfflfflffl}B

s02

0 0 b bd=2 bc=2 0 0

0 0 a ad=2 ac=2 0 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

s03

264 375 12Ae Bs01 Bs02 Bs03 41

Bs1

1

2Ae

0 0 0 1 0 1 0

0 0 0 0 1 0 1

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

s11

0 0 0 1 0 1 0

0 0 0 0 1 0 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

s12

0 0 0 1 0 1 0

0 0 0 0 1 0 1|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B

s13

266664377775 12Ae Bs11 Bs12 Bs13 42


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in which a, b, c, d arecomputed from the coordinates of three nodes of

theelement as shown in Fig. 5, andAe is thearea of thetriangular ele-

mentXe.

Substituting Eqs. (38)(42) into Eq. (27), the global stiffness ma-

trix now becomes

KDSG3

XNe

e1

KDSG3e 43

where KDSG3e is the element stiffness matrix of the DSG3 element,

and is computed by

KDSG3e

ZXe

BTi D

BjdX

ZXe

STi D

s SjdX B

Ti D

BjAe S

Ti D

s SjAe

44

From Eqs. (38)(42) and (44), it is seen that the element stiff-

ness matrix in the DSG3 depends on the sequence of node numbers

of elements, and hence the solution of DSG3 is influenced when the

sequence of node numbers of elements changes, especially for the

coarse and distorted meshes. The CS-DSG3 is hence proposed to

overcome this drawback and also to improve the accuracy as well

as the stability of the DSG3.

5.2. Formulation of CS-DSG3 based on the C0-type HSDT

In the CS-DSG3 [54] using the C0-type HSDT formulation [36],

the domain discretization is the same as that of the DSG3 using

Nn nodes and Ne triangular elements. However in the formulation

of the CS-DSG3, each triangular element Xe is further divided into

three sub-trianglesD1,D2 andD3 by connecting the central point O

of the element to three field nodes as shown in Fig. 6.

In the CS-DSG3, we assume that the displacement vector deO at

the central point O is the simple average of three displacement vec-

tors de1; d

e2 and d

e3 of three field nodes

deO 13de1 d

e2 d

e3

45On the first sub-triangleD1(triangle O-1-2), a linear approxima-

tion ueD1 ueD1 veD1 weD1 beD1x beD1

y /eD1

x /eD1

y

h iTis con-

structed by

ueD1 NeD11 deO N

eD12 d

e1 N

eD13 d

e2 N

eD1 deD1 46

where deD1 d

eO d

e1 d

e2

Tis the vector of nodal degrees of free-

dom of the sub-triangle D1 and NeD1 NeD11 N

eD12 N

eD13

contains

the linear shape functions created by the sub-triangle D1.

Using the DSG3 formulation based on the C0-type HSDT pre-

sented in Section 5.1 for the sub-triangle D1, the membrane strain

eD10 , the bending strains jD11 ; jD12 and the shear strains eD1s ; jD1s inthe sub-triangle D1 are then obtained, respectively, by

eD10 bmD11 b

mD12 b

mD13

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}bmD1

deO

de1

de2

26643775 bmD1 deD1 47

jeD11 bb1D11 b

b1D12 b

b1D13

h i|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

bb1D1

deO

de1

de2

264375 bb1D1 deD1 ;

jeD12 bb2D11 b

b2D12 b

b2D13h i|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}bb2D1

deO

de1

d

e2

264

375

bb2D1 d

eD1

48

eeD1s bs0D11 b

s0D12 b

s0D13

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}b

s0D1

deO

de1

de2

264375 bs0D1 deD1 ;

jeD1s bs1D11 b

s1D12 b

s1D13

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}b

s1D1

deO

de1

de2

264375 bs1D1 deD1

49

where bmD1 ; b

b1D1 ; bb2D1 ; b

s0D1 and bs1D1 are, respectively, com-

puted similarly as the matrices Bm; Bb1 ; Bb2 ; Bs0 and Bs1 of the

DSG3 in Eqs. (38)(42) but with two following changes: 1) the coor-

dinates of three nodexi xi yi T; i 1; 2; 3 are replaced byxO, x1

andx2, respectively; and 2) the areaAe is replaced by the areaAD1 ofsub-triangle D1.

Substituting deO in Eq. (45) into Eqs. (47)(49), and then rear-

ranging we obtain

eD10 13

bmD11 b

mD12

13

bmD11 b

mD13

13

bmD11

h i|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

BmD1

de1

de2

de3

264375 BmD1 de

50

jD11 13

bb1D11 b

b1D12

13

bb1D11 b

b1D13

13

bb1D11

h i|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflffl}Bb1D1

de1

de2

de3

264

375

Bb1D1 de

jD12 13

bb2D11 b

b2D12

13

bb2D11 b

b2D13

13

bb2D11

h i|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflffl}

Bb2D1

de1

de2

de3

264 375 Bb2D1 de51

eD1s 13

bs0D11 b

s0D12

13

bs0D11 b

s0D14

13

bs0D11

h i|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl fflfflfflfflfflfflfflfflffl}

Bs0D1

de1

de2

de3

264375 Bs0D1 de

jD1s 13

bs1D11 b

s1D12

13

bs1D11 b

s1D13

13

bs1D11

h i

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Bs1D1

de1

de2

de3

264375 Bs1D1 de

52

Fig. 5. Three-node triangular element and local coordinates in the DSG3.

Fig. 6. Three sub-triangles (D1, D2 and D3) created from the triangle 1-2-3 in theCS-DSG3 by connecting the central point O with three field nodes 1, 2 and 3.




7/16

Similarly, by using cyclic permutation, we easily obtain the bending

and shear strains eDj0 ; j

Dj1 ; j

Dj2 ; e

Djs ; j

Djs and matrices

BmDj ; Bb1Dj ; Bb2Dj ; Bs0Dj ; Bs1Dj ; j 2; 3, for the second sub-triangle

D2 (triangle O-2-3) and third sub-triangle D3 (triangle O-3-1),

respectively.

Now, applying the cell-based strain smoothing operation in the

CS-FEM [38,39], the membrane strain eDj0 , bending strains j

Dj1 ; j

Dj2

and shear strainse

Dj

s ; j

Dj

s ;j

1;

2;

3 are, respectively, used to cre-

ate a smoothed membrane strain ~ee0, smoothed bending strains~je1; ~j

e2 and smoothed shear strains ~e

es ; ~j

es on the triangular element

Xe, such as:

~ee0 ZXe

eh0UexdX X3j1

eDj0

ZDj

UexdX 53

~je1 ZXe

jh1UexdX X3j1

jDj1

ZDj

UexdX;

~je2 ZXe

jh2UexdX X3j1

jDj2

ZDj

UexdX 54

~ees ZXeebsUexdX X

3

j1 ZDjeDjs UexdX;

~jes ZXe

jhsUexdX X3j1

ZDj

jDjs UexdX 55

where Ue(x) is a given smoothing function that satisfies the unity

propertyRXeUexdX 1. Using the following Heaviside constant

smoothing function

Uex 1=Ae x 2 Xe

0 x R Xe

&56

where Ae is the area of the triangular element, the smoothed mem-

brane strain ~ee0, the smoothed bending strains ~je1; ~j

e2 and the

smoothed shear strains ~ees ; ~jes in Eqs. (53)(55) become

~ee0 1

Ae X3

j1ADje

Dj0 57

~je1 1

Ae

X3j1

ADjjDj1 ;

~je2 1

Ae

X3j1

ADjjDj2 58

~ees 1

Ae

X3j1

ADjeDjs ; ~j

es

1

Ae

X3j1

ADjjDjs 59

Substituting eDj0L; j

Dj1 ; j

Dj2 ; e

Djs , and j

Djs ; j 1; 2; 3, into Eqs. (57)

(59), the smoothed bending strain ~ee0, the smoothed bending strains~je1; ~j

e2 and the smoothed shear strains ~e

es ; ~j

es are expressed by

~ee0 eBmde; ~je1 eBb1 de; ~je2 eBb2 de; ~ees eBs0 de;

~jes eBs1 d

e60

where eBm; eBb1 ; eBb2 ; eBs0 and eBs1 are the smoothed bending andshear strain gradient matrices given by

eBm 1Ae

X3j1

ADj BmDj ; eBb1 1

Ae

X3j1

ADj Bb1Dj ; eBb2 1

Ae

X3j1

ADj Bb2Dj

eBs0 1Ae

X3j1

ADj Bs0Dj ; eBs1 1

Ae

X3j1

ADj Bs1Dj

61

Therefore the global stiffness matrix of the CS-DSG3 are assembledby

eK XNee1

eKe 62where eKe is the element smoothed stiffness matrix for FGMMindlinplates given by

~Ke

ZXe

eBTi DeBjdX ZXe

~STi Ds~SjdX eBTi DeBjAe ~STi Ds~SjAe 63

in which

eBi

eB

mi

TeB

b1

i

TeB

b2

i

T !;

eSi

eB

s0i

TeB

s1i

T !64

From Eqs. (61) and (63), it is clearly seen that the element stiff-ness matrix in the CS-DSG3 does not depend on the sequence of

node numbers, and hence the solution of CS-DSG3 is unchanged

when the sequence of node numbers changes. In addition, due to

using the cell-based strain smoothing technique in the CS-FEM

[38,39], the CS-DSG3 can help to improve the accuracy as well as

the stability of the DSG3 [54].

6. Numerical results

In this section, various numerical examples are performed to

show the accuracy and stability of the CS-DSG3 compared to re-

sults to several published ones in the literature. The ceramicmetal

FGM plates are used, and their material properties, including

Youngs modulus, Poissons ratio, thermal expansion coefficient,conductivity and density of pure plates are given in Table 1.

6.1. Temperature field distribution in the FGPs subjected to only

surface temperatures

A functionally graded plate shown in Fig. 1 is made of

Aluminum-Ziconia (Al/ZnO2-2). The thermal analysis is conducted

by imposing constant surface temperatures at the Aluminum

(metal) at 20 C and Ziconia (ceramic) surface at 300 C. The varia-

tion of temperature varies only through the thickness direction.

The distribution of temperature through the thickness is attained

by solving the one-dimensional steady state heat conduction

equation, which is shown in Eq. (3). In this analysis, the standard

Table 1

Material properties of FGM components.

Properties Aluminum Alumina

(Al2O3)

Zirconia (ZrO2-

1)

Zirconia (ZrO2-

2)

E (GPa) 70 380 200 151

m 0.3 0.3 0.3 0.3k (W/mK) 204 10.4 2.09 2.09

a (106/C)

23 7.2 10 10

q (kg/m3) 2707 3800 5700 3000Fig. 7. Two square plate models: (a) clamped plate; (b) simply supported plate.


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8/16

Fig. 8. Four lowest frequencies of square plates by the CS-DSG3 corresponding to different values of volume fraction exponents: (a) n = 0; (b) n = 0.5; (c) n = 1; (d) n = 2.

Fig. 9. Effect of volume fraction exponent n to four lowest frequencies of square FGM plates: (a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4.




9/16

one-dimensional finite element method is used.

Fig. 3 illustrates the temperature distributions through the

thickness Aluminum-Ziconia (Al/ZnO2-2) FGM plate for the various

value of volume fraction exponent n. With the results of the tem-

perature through the thickness, the thermal force and moment

resultants Nth, Pth and Mth can be computed according to Eq. (20).

6.2. Free vibration analysis of FGPs

In this section, we investigate the accuracy and efficiency of theCS-DSG3 element for analyzing natural frequencies of plates. The

plate may have simply (S) supported or clamped (C) edges. The

symbol, CCCC or SSSS, for instance, represents clamped or simply

supportted boundary conditions along the four edges of the

rectangular plate, respectively. A non-dimensional frequency

parameter - is often used to stand for the frequencies. The numer-ical results by the CS-DSG3 are then compared to analytical solu-

tions and other available numerical results in the literature.

6.2.1. Square FGM plates

We now consider two square Al/Al203 plates of length L = 10and thickness t= 1 with two boundary conditions, CCCC and SSSS,

Fig. 10. Shape of six lowest eigenmodes of a clamped square FGM plate by the CS-DSG3: (a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4; (e) Mode 5; (f) Mode 6.

Fig. 11. (a) a skew plate model; (b) a discretization of skew plate using triangular elements.


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10/16

Fig. 12. Four lowest frequencies of the Al/ZrO2-2 skew plates with h = 15: (a) n = 0; (b) n = 0.5; (c) n = 1; (d) n = 3.

Fig. 13. Four lowest frequencies of skew plates with h = 45: (a) n = 0; (b) n = 0.5; (c) n = 1; (d) n = 3.




11/16

as shown in Fig. 7a and b. A regular discretization using 144 3-node

triangular elements and a non-dimensional frequency parameter

- xL2=tffiffiffiffiffiffiffiffiffiffiffiffiqc=Ec

pare used. Fig. 8 shows the four first non-dimen-

sional frequencies of the FGM plate by the CS-DSG3 corresponding

to different values of volume fraction exponents, n = 0, n = 0.5, n = 1

and n = 2. It is seen that the present results match well with those

ofkp-Ritz [33].

Next, the effect of volume fraction exponent n to the non-

dimensional frequency of Al/Al203 plate is investigated and the re-

sults are shown in Fig. 9. Again, it is seen that the present results

match well with those by kp-Ritz [33]. Fig. 10 plots the shape of

six lowest eigenmodes of the clamped square FGM plates by the

CS-DSG3. It is observed that the shapes of eigenmodes express cor-rectly the real physical modes of the plate.

Fig. 14. Shape of six lowest eigenmodes of a clamped skew plate by the CS-DSG3: (a) Mode 1; (b) Mode 2; (c) Mode 3; (d) Mode 4; (e) Mode 5; (f) Mode 6.

Table 2

Non-dimensional center deflection w 100wcEmt3=121 m2qL4 of a Al/ZrO2-1

plate (L/t= 5) subjected to a uniform load.

Boundary condition Method n

0 0.5 1 2

CCCC DSG3 0.0731 0.0970 0.1137 0.1334

CS-DSG3 0.0732 0.0972 0.1139 0.1336

MLPG 0.0718 0.1073 0.1253 0.1444

kp-Ritz 0.0774 0.1034 0.1207 0.1404

SSSS DSG3 0.1768 0.2336 0.2679 0.3051

CS-DSG3 0.1774 0.2343 0.2688 0.3060

MLPG 0.1671 0.2505 0.2505 0.3280

kp-Ritz 0.1722 0.2403 0.2403 0.3221

Fig. 15. Non-dimensional central deflection w 100wcEmt3=121 m2qL4

of asimply supported FGM plate subjected to a uniform load by the CS-DSG3.

Fig. 16. Non-dimensional central deflection w 100wcEmt3=121 m2qL4 of a

clamped FGM plate subjected to a uniform load by the CS-DSG3.




12/16

6.2.2. Skew FGM plates

We now consider a Al/ZrO2-2 skew plate with various skew an-

gles h as shown in Fig. 11a. Two boundary conditions (SSSS and

CCCC) are studied, and the plate is discretized into 144 three-node

triangular elements as shown in Fig. 11b.

Figs. 12 and 13 show the first four non-dimensional frequencies

of the Al/ZrO2-2 skew plate with skew angles h = 15 and h = 45 by

the CS-DSG3, corresponding four different values of volume frac-

tion exponents, n = 0, 0.5, 1.0 and 3.0. It is seen that the results

by the CS-DSG3 agree well with those by kp-Ritz [33]. Furthermore,

Fig. 14 plots the shape of six lowest eigenmodes of the clamped

skew plate with skew angle h = 60 by the CS-DSG3. It is again ob-

served that the shapes of eigenmodes express correctly the real

physical modes of plate.

6.3. Static analysis

6.3.1. Square FGM plates

6.3.1.1. Behavior of FGM plates subjected to only the mechanical

load. We now perform a static analysis for two square Al/ZrO2-1

plates (length L, thickness t) with clamped and simply supported

boundary conditions as described in Fig. 7a and b. The plate is sub-

jected to a uniform load q = 1 and the volume fraction exponent n

ranges from 0 to 2. Using the uniform discretization of 13 13

nodes, the non-dimensional center deflectionsw 100wcEmt

3=121 m2qL4 of the plate (L/t= 5) by the CS-DSG3 are presented in Table 2, and compared with those of Mesh-

less Local Petrov Galerkin (MLPG) [31] and kp-Ritz [11]. It is seen

that the numerical results by the CS-DSG3 agree well to reference

solutions. In addition, the ability of overcoming the shear locking

of the CS-DSG3 for the cases of FGM plates with different boundary

conditions is illustrated clearly in Figs. 15 and 16. It isseen thatthe

non-dimensional central deflection of the FGM plate is almost

independent with the change of the length-to-thickness ratio L/t

when the plates become thin. The conclusion is correct for the

plates with two simply supported and clamped boundary condi-

tions, and also for four different volume fraction exponents, n = 0,

0.5, 1.0 and 2.0.

Next, Fig. 17 shows the effect of volume fraction exponent n to

the deflection of the center line position of simply supported FGM

plate by the CS-DSG3. It is seen that when the volume fraction

Fig. 17. Effect of volume fraction exponent n to the non-dimentional deflectionw wcEmt

3=121 m2qL4 of the center line position of a simply supported FGMplate by the CS-DSG3.

Fig. 18. Non-dimentional axial stress profile rx rxt2=qL2

through thethickness of a simplysupported FGM plate subjectedto a uniformload by the CS-DSG3: (a) n = 0;(b)n = 0.5; (c) n = 1; (d) n = 2.




13/16

exponent n increases, the deflection also increases, as expected.

This result is quite similar to those by MLPG [31] and kp-Ritz [11].

6.3.1.2. Behavior of FGM plates subjected to the thermo-mechanical

load. In this example, the proposed CS-DSG3 is used to analyse the

behavior of the FGPs subjected to both mechanical and thermal

loadings. The temperature is assumed to be constant in the plane

of the plate and varies through the thickness. The FGM plate (Al/

ZrO2-2) is made from aluminum at the bottom surface and the Zir-conia at the top surface with the length L = 0.2 m and thickness

t= 0.01 m. Material properties vary through the thickness of plate

with respect to the power law as shown in Eqs. (1) and (2).

We first investigate the behavior of the FGM plate subjected to a

uniform mechanical load q, and the volume fraction exponents n of

the plate are varied by n = 0, 0.5, 1 and 2. Figs. 18 and 19 show the

axial stress profile through the thickness of FGM plates

corresponding to two boundary conditions, simply supported and

clamped ones. It is seen that the present results by the CS-DSG3

match well with those by MLPG [31]. Fig. 20 shows the effect ofthe volume fraction exponent and the mechanical load parameter

Fig. 19. Non-dimensional axial stress profile rx rxt2=qL2 through the thickness of a clamped FGM plate subjected to a uniform load by the CS-DSG3: (a) n = 0; (b) n = 0.5;

(c) n = 1; (d) n = 2.

Fig. 20. Effect of the volume fraction exponent and the mechanical load parameter

P qL4=Emt4 (without thermal load) to non-dimensional central deflection

wc wc=t of a simply supported FGM plate by the CS-DSG3.

Fig. 21. Effect of the volume fraction exponent and the thermal load Tc to non-

dimensional central deflection wc wc=t of a simply supported FGM plate by the

CS-DSG3.




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Fig. 22. Effect of the volume fraction exponent and the mechanical load parameter

P qL4=Emt4 (with thermal loads, Tm = 20 C and Tc = 300 C) to non-dimen-

sional central deflection wc wc=t of a simply supported FGM plate by the CS-

DSG3.

Fig. 23. Non-dimensional deflection wc wc=t of the central line position of a

simply supported FGM plate subjected to a mechanical, thermal and thermo-

mechanical loads by the CS-DSG3.

Fig. 24. Effect of the thermal load Tc on central deflection wc wc=t of a simply

supported FGM plate (with n = 1) subjected to a thermo-mechanical load by the CS-DSG3.

Fig. 25. Effect of the skew angle on the axial stress through the thickness of a

simply supported Al/ZnO2-1 plate (n = 0.5) subjected to a mechanical load by the

CS-DSG3.


simply supported Al/ZnO2-1 plate (n = 2) subjected to a mechanical load by the CS-

DSG3.


simply supported Al/ZnO2-1 plate (n = 0.5) subjected to a thermo-mechanical loadby the CS-DSG3.




15/16

P qL4=Emt4 to non-dimensional central deflection wc wc=t

of the simply supported FGM plate. It is observed that the central

deflection of the plate varies linearly with respect to the change

of the load parameter. In addition as expected, the central deflec-

tion increases correspondingly to the increase of the volume frac-

tion exponent n. This is because a FGM plate with a larger volume

fraction exponent means that it has a smaller ratio of ceramic com-

ponent, and hence its stiffness is reduced.

We next study the behavior of the FGM plate subjected to the

thermal load. The temperature at the bottom surface is hold at

Tm = 20 C and the temperature at the top surface is varied from

0 C to500 C. Fig. 21 shows the effect of the volume fraction expo-

nent and the thermal load to non-dimensional central deflection

wc wc=t of the simply supported FGM plate. It is observed thatthe central deflection of the plate also varies linearly with respect

to the change of the thermal load. In addition, when the thermal

load increases highly at the top surface, the deflection of the plate

increases upwardly. This implies that the FGM plates subjected to

mechanical loads in the high temperature condition will have the

deflection smaller than that of the FGM plates subjected to only

mechanical loads in the normal temperature conditions. In other

words, the FGM plates can resist mechanical loads in the high tem-

perature conditions very well, as illustrated in the following

numerical results.

We nowinvestigate the behavior of FGM plates subjected to the

thermo-mechanical load. The temperature is now holdTm = 20 C at

the bottom surface and Tc = 300 C at the top surface. Fig. 22 shows

the effect of the volume fraction exponent and the mechanical load

parameter

P qL

4

=Emt

4

(from

14 to 0) to non-dimensional cen-

tral deflection wc wc=t of the simply supported FGM plate. It is

seen that the behavior of deflection under thermo-mechanical load

is quitedifferent fromthe pure mechanical load. When the mechan-

ical load parameter equals to zero,the deflection of FGMplate is dif-

ferent from zero. This is because the high temperature at the top

surface causes the thermal expansion, and hence makes the plates

deflect upwardly. In addition as expected, the central deflection in-

creasescorrespondinglyto the increaseof the volumefractionexpo-

nent. Next, Fig. 23 shows the non-dimensional deflection of the

central line position of a simply supported FGM plate subjected to

a mechanical, thermal and thermo-mechanical loads. As expected,

it is observedthat when theplate is subjectedto the themo-mechan-

ical load, thedeflection of plate is smaller than that of the plate sub-

jected to onlythe mechanical load. Lastly, Fig. 24 shows theeffect of

the temperature Tc on the central deflection of a simply supported

FGM plate subjected to a thermo-mechanical load and the volume

fraction exponent n = 1. Again, it is seen that when the temperature

Tcat the topsurfaceof FGMplate increases, the upwarddeflection by

the thermal load is also increased and hence the total deflection of

FGM plate becomes smaller than those of the FGM plate subjected

to only pure mechanical loads.

6.3.2. Skew FGM plate

Wenow applytheproposedCS-DSG3to analysea simplysupported

Al/ZrO2-1 skewplate as shown in Fig. 11a. The geometry of plate is gi-

ven by length L = 10, thickness t= 0.1 and the plate is discretized by a

uniformmesh of 13 13 nodes as shown in Fig. 11b. Wefirstconsider

the effect of the skew angleh

to the axial stress of the FGM plate sub-

jectedto a uniformmechanical load q = 104 N/m2. The resultsfor three

volume fraction exponents at n = 0.5 and n = 2.0 are shown in Figs. 25

and 26, respectively. It is seen that when the skew angle becomes lar-

ger, the absolute axial stress induced in the plate becomes higher.

These results agree very well with those given in the reference [11].

A similar observation for thecase of the FGM plate subjected to a ther-

mo-mechanical load is also illustrated in Figs. 27and28. Lastly, Fig. 29

shows the effect of the temperature on the axial stress through the

thickness of the FGM plate subjected to a thermo-mechanical load at

the volume fraction exponent n = 1. It is again observed that when

the temperature becomes larger, the absolute axial stress induced

in the plate becomes higher.

7. Conclusions

The paper presents an extension of the CS-DSG3 using three-

node triangular elements to the C0-type high-order shear deforma-

tion plate theory for the static and free vibration analyses of func-

tionally graded plates (FGPs). In the FGPs, the material properties

are assumed to vary through the thickness by a simple power rule

of the volume fractions of the constituents. In the static analysis,

both thermal and mechanical loads are considered and a two-step

procedure is performed including a step of analyzing the tempera-

ture field along the thickness of the plate and a step of analyzing

the behavior of the plate subjected to both thermal and mechanical

loads. The accuracy and reliability of the proposed CS-DSG3 is ver-

ified by comparing its numerical solutions with those of other

available numerical results. In addition, the numerical results bythe CS-DSG3 confirm that the FGM plates can resist mechanical


simply supported Al/ZnO2-1 plate (n = 2.0) subjected to a thermo-mechanical load

by the CS-DSG3.

Fig. 29. Effect of the thermal load Tc on the axial stress through the thickness of a

simply supported Al/ZnO2-1 plate (n = 1.0) subjected to a thermo-mechanical loadby the CS-DSG3


http://dx.doi.org/10.1016/j.commatsci.2013.06.010http://dx.doi.org/10.1016/j.commatsci.2013.06.010http://-/?-http://-/?-


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loads in the high temperature conditions very well. This is due to

the opposite responses of the FGM plate to the thermal loads and

mechanical loads.

Acknowledgements

This work was supported by Vietnam National Foundation for

Science & Technology Development (NAFOSTED), Ministry of Sci-

ence & Technology, under the basic research program (Project

No.: 107.02-2012.05).

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7. a cell-based smoothed discrete shear gap method (cs-dsg3) based on the c0 type higher order shear...

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