FGMnol

Accepted Manuscript

Non-linear analysis of FGM plates under pressure loads using the higher-order

shear deformation theories

R. Sarfaraz Khabbaz, B. Dehghan Manshadi, A. Abedian

PII: S0263-8223(08)00205-5

DOI: 10.1016/j.compstruct.2008.06.009

Reference: COST 3460

To appear in: Composite Structures

Please cite this article as: Khabbaz, R.S., Manshadi, B.D., Abedian, A., Non-linear analysis of FGM plates under

pressure loads using the higher-order shear deformation theories, Composite Structures (2008), doi: 10.1016/

j.compstruct.2008.06.009

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http://dx.doi.org/10.1016/j.compstruct.2008.06.009



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Non-linear analysis of FGM plates under pressure loads using the

higher-order shear deformation theories

R. Sarfaraz Khabbaza, B. Dehghan Manshadia, A. Abediana,*

a Department of Aerospace Engineering, Sharif University of Technology, Tehran, P.O. Box 11365-4563, Tehran, Iran

*) Corresponding Author

Tel: +98 21 66164947

Fax: +98 21 66022731

E-mail: [email protected]

Abstract

In this study the energy concept along with the first and third order shear deformation

theories (FSDT and TSDT) are used to predict the large deflection and through the

thickness stress of FGM plates. These responses are studied and discussed as a function of

plate thickness and the order " n " of a power law function which is considered for the

through the thickness variation of the properties of the FGM plate. The results show that the

energy method powered by the FSDT and FSDT is capable of predicting the effects of plate

thickness on the deformation and the through the thickness stress. Here, also the effects of

power " n " on the plate response is clearly depicted. Notably, the singularity of the stress

distribution for very small and very large " n " values is demonstrated.

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Keywords: Large deformation; Functionally graded material; Shear deformation theory;

Energy method.

1. Introduction

Traditional composites which are usually composed of two different materials have been

broadly used to fulfill the increasing high performance industrial demands. However, due to

discontinuity of material properties at the interface of composite constituents, the stress

fields in this region under some loading conditions such as high-temperature environment

show some kind of singularity. For example, in the combustion chamber of air vehicle

engines or a nuclear fusion reaction container, the relatively higher mismatch in thermal

expansion coefficients of constituent materials will induce high residual stresses which may

consequently lead to cracking or debonding. To eliminate the stress singularities in ultra-

high-temperature environments, the concept of functionally graded materials (FGMs) was

first introduced in 1984 by a group of material scientists in Japan to [1, 2].

In FGMs, which are microscopically inhomogeneous and assumed to be a kind of

composite material, the mechanical properties vary smoothly and continuously from one

surface to the other. This is achieved by gradually varying the volume fraction of the

constituent materials. By incorporating the variety of possibilities inherent with the FGM

concept, new property functions are tailored and the materials performance in harsh

environments could be improved. In this regard, the FGMs were initially designed as

thermal barrier materials for aerospace structural applications and fusion reactors and

nowadays are developed for a more general use as structural components in extremely high-

temperature environments.

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In the simplest FGMs, the volume percentage (or volume fraction, fV ) of the constituent

materials change gradually from one surface to the other. In some FGMs discontinuous

changes such as a stepwise gradation of the material ingredients is considered. The most

familiar FGM is compositionally graded from a refractory ceramic to a metal substance.

The ceramic in a FGM offers thermal barrier effects and protects the metal from corrosion

and oxidation, and the FGM is toughened and strengthened by the metallic composition. A

mixture of ceramic and metal with a continuously varying volume fraction can be easily

manufactured.

A wide range of results on linear behavior of functionally graded plates with different

material function models are available in the literature [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,

and 15]. In these studies various plate theories such as classical or higher order shear

theories have been used. However, nonlinear investigations of FGM plates under

mechanical loading are limited in number. For example, the Poincare method was used [16]

to examine the thermally induced large deflection of a simply supported, FGM thin plate

with the Young’s modulus symmetrically varying up to the plate mid-plane passing through

the thickness. Also, elastic bifurcation buckling of FGM plates under in-plane compressive

loading using a combination of micromechanical and structural approaches has been

reported [17]. A comprehensive FEM analysis of nonlinear static and dynamic response of

functionally graded ceramic–metal plates subjected to simultaneous thermal and transverse

mechanical loads using first order shear deformation plate theory would be found in [18].

Moreover, using Karman theory for large deformation, the results of analytical solution for

plates and shells under transverse mechanical loads and a temperature field have been

presented in [19]. In the follow up study, the large deflection and post-buckling response of

functionally graded rectangular plates under transverse and in-plane loads using a semi-

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analytical approach have been conducted [20]. In addition, post buckling analysis of FGM

plates with piezoelectric actuators under thermo-electro-mechanical loadings has been

reported in [21]. Also, the mentioned study presents post buckling analysis of a simply

supported and shear deformable functionally graded plate with piezoelectric actuators

subjected to simultaneous application of mechanical, electrical, and thermal loads.

Additionally, the post buckling of the axially loaded FGM hybrid cylindrical shells with

piezoelectric actuators subjected to axial compression combined with electric loads in

thermal environments has been studied in [22]. Besides, relationships between

axisymmetric bending and buckling response of FGM circular plates based on third-order

shear deformation plate theory (TSDT) and classical plate theory (CPT) have been

presented in [23]. The results of a study on the nonlinear effects of geometry on static and

dynamic responses of isotropic, composite, and FGM beams using the new beam element

(which was introduced by Chakraborty et al.) could be found in [24]. Most recently, a study

on large deformation behavior of functionally graded plates subjected to pressure loads has

been conducted using energy concept [25]. In this study, the material properties have been

assumed to be distributed in the thickness direction according to a simple power law

function in terms of volume fractions of the constituents. As for the boundary condition, the

plate was considered to be simply supported on its all four edges. The constitutive

equations for rectangular plates of FGM were obtained using the Von-Karman theory for

large deflections and the solution was obtained by minimization of the total potential

energy.

In the present study, large deflection behavior of a simply supported elastic rectangular

FGM plate subjected to a pressure loading is investigated. The material properties of the

FGM plate, except for the Poisson’s ratio which is constant, are assumed to vary

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continuously throughout the thickness of the plate. The variations are considered to be in

accordance with the volume fraction of the constituent materials based on a power-law

function. In this paper, assuming sinusoidal deflections, the problem is solved using the

first- and third-order shear deformation theories. The solutions are achieved by minimizing

the total potential energy and the results are compared to the classical plate theory.

2. Formulation

2.1. Properties of the FGM constituent materials

A FGM can be defined by varying the volume fractions of the constituent materials through

a function. Several available analytical and computational studies have discussed the issue

of finding suitable functions considering some selection criteria. The function must be

continuous, simple, and have the ability to exhibit curvatures of both ‘‘concave upward’’

and ‘‘concave downward’’ [26]. In this research work, FGM plates with different

thicknesses with a power-law function are considered. The configuration of elastic

rectangular plates is considered as shown in Fig. 1. The material properties i.e. Young's

modulus ( )E and the Poisson's ratio ( )υ , are normally considered to be varied from upper to

the lower surface of the plate such that the top surface (i.e. 2hz += ) is ceramic-rich,

whereas the bottom surface (i.e. 2hz −= ) is metal-rich. However, it is known that the

effect of Poisson's ratio on the deformation is much less than that of Young's modulus [27].

Thus, υ of the plates is assumed to be constant. Therefore, here the young's modulus of the

plates is assumed to vary continuously only in the thickness direction ( z -axis), i.e.

)z(EE = and )z(υ=υ , only.

The volume fraction ( (z)ϑ ) of the plates is assumed to be a power-law function (P-FGM),

i.e.

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n)

h2hz

((z)+=ϑ (1)

where n is the material parameter and h is the thickness of the plate. Besides, the material

properties of a P-FGM can be determined by the rule of mixture [28]

21 (z)]E[1(z)EE(z) ϑϑ −+= (2)

where E(z) indicates a typical Young's modulus and 1E and 2E denote the Young's

modulus of the bottom (i.e. 2hz += ) and top surface (i.e. 2hz −= ), respectively. The

variation of Young's modulus in the thickness direction of the P-FGM plate for different n

values is depicted in Fig. 2 the figure shows that for 1>n the E(z) changes rapidly near

the bottom surface while it increases quickly near the top surface for 1<n .

2.2. Fundamental Equations of rectangular FGM plates

A linearly-elastic rectangular FGM plate subjected to a pressure load is considered. The

Classical Plate Theory (CPT), First- and Third-order Shear Deformation Theories (FSDT

and TSDT) are applied throughout this work. The general displacement field for CPT,

FSDT, and TSDT can be written as [29]

),()(),(

)(),(),,(~ yxzgx

yxwzfyxuzyxu xφ+

∂∂+=

),()(),(

)(),(),,(~ yxzgy

yxwzfyxvzyxv yφ+

∂∂+=

),(),,(~ yxwzyxw = (3)

where (u~ , v~ , w~ ) are the displacements corresponding to the co-ordinate system and are

functions of the spatial co-ordinates; (u , v , w ) are the displacements along the respective

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axes of x , y , and z , and xφ and - yφ are the rotations about y and x-axes. Note that for

each one of the theories considered the functions )(zf and )(zg are defined as below

i) for the CPT:

==

0)()(

zg

zzf ,

ii) for the FSDT:

==

zzg

zf

)(0)(

, and

iii) for the TSDT:

−=

=

)3

4()(

)3

4()(

23

23

hzzzg

hzzf

The CPT's displacement field implies that straight lines normal to the xy plane before and

after deformation remain straight and normal to the plate mid-surface. The Kirchhoff

assumption amounts to neglecting both transverse shear and transverse normal effects, i.e.

deformation is due entirely to bending and in-plane stretching. On the other hand, the FSDT

extends the kinematics of the classical plate theory by including a gross transverse shear-

deformation in its assumptions, i.e. the transverse shear strain is assumed to be constant

with respect to the thickness coordinate. In this theory, shear correction factors are

introduced to correct for the discrepancy between the actual transverse shear-force

distributions and those computed using the kinematics relations of FSDT. However, the

assumed relationships for TSDT's displacement field accommodate quadratic variation of

transverse shear strains (and hence stresses) and vanishing boundary requirement for

transverse shear stresses at the top and bottom surfaces of a plate. Thus, there is no need to

use shear correction factors in a third-order theory. These theories provide some increase in

accuracy relative to the FSDT solution at the expense of a significant increase in the

computational efforts.

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According to the nonlinear strain-displacement relationships [30], the strain terms

compatible with the displacement field of Eq. (3) are

ηψεε )()(~ zgzf ++= (4a)

where

=

zx

yz

xy

y

x

γγγεε

ε

~~~~~

~ ,

∂∂∂∂

∂∂

∂∂+

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

=

xwyw

yw

xw

xv

yu

yw

yv

xw

xu

2

2

)(21

)(21

ε ,

∂∂

∂∂

∂∂

∂∂

∂∂∂−

∂∂−

∂∂−

=

xw

zzf

zf

yw

zzf

zf

yxw

yw

xw

)()(

1

)()(

1

22

2

2

2

2

ψ ,

and

∂∂

∂∂

∂∂

+∂

∂∂

∂∂

∂

=

y

x

yx

y

x

zzg

zg

zzg

zg

xy

y

x

φ

φ

φφ

φ

φ

η

)()(

1

)()(

1

(4b)��

Also, the stress–strain relationships in coordinate system of the FGM plate can be

expressed as

=

zx

yz

xy

y

x

zx

yz

xy

y

x

Q

Q

Q

QQ

QQ

γγγεε

τττσσ

~~~~~

000000000000

000000

~~~~~

33

33

33

2221

1211

(5-a)

where

22211 1)(

υ−== zE

QQ , 22112 1)z(E

QQυ−

υ== , )1(2

)(33 υ+

= zEQ (5-b)

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The stress and moment resultants of the FGM plate can be obtained by integrating Eq. (5)

over the thickness, and are written as

∫−=

2

21

h/

h/ ijijij dz,z)�(),M(N (6)

where i and j stand for x and y and hence

×

=

}{}{}{

][][][][][][

}{}{

222

111

ηψε

CBA

CBA

M

N (7)

in which ][ kA , ][ kB , and ][ kC are given by

∫−=

2/

2/21 ).,1(),(h

h ijijij dzQzAA (8-a)

∫−=

2/

2/21 ).().,1(),(h

h ijijij dzQzfzBB (8-b)

∫−=

2/

2/21 ).().,1(),(h

h ijijij dzQzgzCC (8-c)

3. Solution

The total potential energy ( Π ) of the FGM plate is determined by summation of strain

energy and the change in potential energy of the uniform externally applied pressure and is

written as

VU +=Π (9)

Here, V (the potential energy of uniform pressure) is given by

∫ ∫=a a

dxdyyxqwV0 0

),( (10)

where q is the uniformly distributed load and the integral limit a is the projected length of

FGM plate in xy plane as shown in Fig. 1. Also, the strain energy (U ) is defined as

dzdxdyUa a h

h

T∫ ∫ ∫−=

0 0

2/

2/

~~21 εσ (11)

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By substituting Eqs. (4) and (5) into Eq. (11) the strain energy finds the following form

dzdxdyQgfQgQf

dzdxdyQgQfQU

a a h

h

TTT

a a h

h

TTT

∫ ∫ ∫

∫ ∫ ∫

−

−

+++

++=

0 0

2/

2/

0 0

2/

2/

22

}].[.2].[2].[2{21

}].[].[][{21

ηψηεψε

ηηψψεε (12)

Now considering the boundary condition for the simply supported FGM plate as in Eq. (1),

the principal of minimum potential energy is applied assuming a first guess solution for the

considered displacements and rotations (i.e. u , v , w , xφ and yφ ) over the mid-surface of

the plate as in Eq. (13).

0),()0,(),(),0(

0),()0,(),(),0(0),()0,(),(),0(

0),()0,(),(),0(0),()0,(),(),0(

====

========

========

axxyay

axMxMyaMyM

axwxwyawyw

axvxvyavyv

axuxuyauyu

xxyy

xxyy

φφφφ

(13)

The required mentioned displacement and rotation fields, which satisfy the simply

supported boundary conditions, are defined as in Eq. (14) [30]

)sin().2

sin(.),(ay

ax

cyxuππ=

)2

sin().sin(.),(a

yax

cyxvππ=

)sin().sin(.),(ay

ax

wyxwππ

�=

)sin().cos(.),(ay

ax

yxx

ππφφ�

=

)cos().sin(.),(ay

ax

yxy

ππφφ�

= (14)

where c , �

w , and �

φ are arbitrary parameters and are determined minimizing the total

potential energy as given in Eq. (15)

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0),,(

=∂

∂

�φ

Πcwc

(15)

Eq. 15 provides a set of three nonlinear equilibrium equations in terms of c , �

w , and �

φ

which should be solved. The obtained constants are then used to calculate the

displacements and rotations (Eqs. (14)) and subsequently the strain and stresses are found

using Eqs. (4) and (5).

4. Results and Discussion

To examine the proposed solution to the FGM problems, a plate consisting of aluminum

and alumina as the respective metal and ceramic substances of a FGM is considered as an

example. Young's modulus for aluminum is 70 GPa while for alumina it is 380 GPa. Note

that the Poisson's ratio is selected constant and equal to 0.3 for both of the constituents. For

all analyses, the lower surface of the plate is assumed to be rich in metal (aluminum) and

the upper surface to be rich in ceramic (alumina). Also, as was previously mentioned, the

volume fraction ( (z)ϑ ) of the FGM plate is assumed to be varied through the thickness with

a power-law function. Hence, according to Eq. 2, the variation of Young's modulus in the

thickness direction is as depicted in Fig. 2. For the results presented in this section the

proposed analytical model is verified by comparing deflection of the plate center point by

the existing results in [25], first. Then a short explanation for the reasons of using FSDT

and TSDT is offered. Finally through the thickness stress of the FGM plate for three

thicknesses of ( h =0.01, 0.05, and 0.1 m ) and assuming different " n " values for the power

law function (e.g. n =100, 2, 1, 0.5, 0.01, and 0) are presented and discussed.

The analytical results are presented in terms of dimensionless deflection and stress. The

dimensionless parameters used here are as follows [19]

aspect ratioha

AR =

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dimensionless centre deflection awW /=

load parameter )/( 41

4 hEqaQ =

dimensionless axial stress )/( 21

2 hEaxσσ =

dimensionless thickness coordinate hzZ /= (18)

Note also that the analyses are performed on a square plate of side mma 200= and

thickness mmh 10= with simply supported boundary condition.

Fig. 3 shows the dimensionless deflection of center of the plate with aspect ratio of

= 20

ha

. As it is seen, the obtained results here match the reported results in [25],

perfectly. Note the later results were obtained using the classical plate theory. Also, based

on the figure, the results obtained by FSDT and TSDT coincide with the results of CPT.

This is well explained by the large plate aspect ratio

= 20

ha

or the small plate thickness

( )01.0=h . For such a plate the in-plane shear stress due to the thin thickness is negligible

and as a result all the applied theories end up with similar results. This is also clearly shown

in Fig. 4 which presents the inverse of the dimensionless deflection

wa

of the plate as a

function of the plate aspect ratio

ha

. Based on the figure, the difference between the

applied theories show up for lower aspect ratios or higher plate thicknesses. For

<10

ha

, it

is seen that highest deflection is predicted by the TSDT followed by the FSDT and finally

the CPT predicts the lowest deflection for the plate. One last point here is that the

mentioned theories remain unaffected by the power " n " for thin FGM plates (see Fig. 3).

However, it is not the case for the through the thickness stress which will be discussed later.

To show the effect of power " n " on deflection of thicker plates the calculations are

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repeated for 1000 ≤≤ n and 1.0=h . Fig. 5 shows the obtained results by the applied

theories. As it is seen, for small " n ", the plate will be rich in ceramic (alumina), which has

a large Young's modulus, and as a result its deflection will be small. Also, base on the

figure, the CPT predicts lower deflection than the other two theories and the predictions by

FSDT and TSDT are close, though the latter gives the largest deflection for the plate.

As for the through the thickness stress of the plate, the results are presented for different

values of n =100, 2, 1, 0.05, 0.01, and 0. Note for each one of " n " values, three different

thicknesses of h =0.01, 0.05, and 0.1 are considered. As Fig. 6 shows, a large stress builds

up in a pure ceramic plate (i.e. 0=n ). Where, with increasing the plate thickness, the in-

plane shear causes slipping of the material layers resulting in dissipation of some of the

system energy. This energy lost is picked up well by the FSDT and TSDT. The deflections

calculated by these theories are 10-15 percent lower than the CPT, see Fig. 6. Repeating the

calculations for the case of pure metal (aluminum or 100=n ) plate, some lower stress but

with similar trend is expected. Here, it is assumed that 100=n provides the condition of

pure metal case which mathematically speaking this value of " n " represents a plate which

is mostly metal but extremely small layer of ceramic shows up on the upper surface of the

plate. As Fig. 7 shows, the graph of stress experiences a sharp bent near the upper surface

due to existence of this small trace of ceramic substance. This has not been predicted by the

formulation of FGM problem in [25]. This material discontinuity (i.e. formation of a very

thin layer of ceramic on the top surface) causes a sharp change in the stress value which

some times interpreted as a kind of singularity due to the rapid change in the material

property. As it is seen in the figure, with increasing the plate thickness the calculated stress

by previously mentioned theories does not coincide any more. The compressive nature of

stress at the top surface predicted by the CPT for thicker plates appears to be tensile if one

applies higher order theories. It also worth to note that the stress gradients calculated by the

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FSDT and TSDT show some decrease compared to the CPT. Similar result is noticed in

Fig. 6 for the pure metal case.

Now by decreasing the value of " n " to 2, 1, and 0.5 (see Figs. 8-10) from 100=n , the

gradient and value of the stress show some changes compared to the results shown in Fig.

7, but the CPT results match well with the results published in [25]. Interestingly for n =2,

the stress calculated by the FSDT and TSDT at the upper surface (which used to be

negative for n =0.5 and n =1) appears to turn tensile for the thick plate with h =0.1 (see

Fig. 8). Fig. 10 presents another interesting result that is the bent seen in the stress graph

near the lower surface of the plate for n =0.5. Note that this value of " n " represents a plate

rich in ceramic and a thin layer of metal at the bottom surface. This could be explained with

the same reasoning as was discussed for Fig. 7 earlier. Fig. 11 which presents the result for

n =0.01 clearly elaborates on the mentioned bent in the stress distribution of Fig. 10. In

fact, the formulation here is capable of picking up the singularities inherent with the joined

dissimilar materials which occur for very small or very large values of " n ".

5. Conclusion

From the analysis performed here the following points could be clearly highlighted:

1. The energy method powered by the FSDT and TSDT could be used for analysis

FGM thick plates.

2. The method is capable of identifying the stress singularities inherent with very small

and very large values of "n".

3. The method shows that significant changes in stress values occur for thick FGM

plates compared to the results of CPT. The compressive stress on the upper surface

changes to tensile with increasing the plate thickness which only the FSDT and

TSDT are able to pick these changes up.

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[26] Markworth AJ, Ramesh KS, Parks WP. Modeling studies applied to functionally

graded materials. Journal of Materials Science 1995;30:2183–2193.

[27] Delale F, Erdogan F. The crack problem for a nonhomogeneous plane. ASME Journal


[28] Bao G, Wang L. Multiple cracking in functionally graded ceramic/metal coatings.

International Journal of Solids and Structure 1995;32:2853–2871

[29] Reddy JN, Wang CM. An overview of the relationships between solutions of the

classical and shear deformation plate theories. Composites Science and Technology

2000;60:2327–2335.

[30] Reddy JN. Mechanics of laminated composite plates. CRC Press, Boca Raton, 1997.

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Fig. 1. The geometry of a typical FGM plate

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19

0

50

100

150

200

250

300

350

400

-0.005 -0.003 -0.001 0.001 0.003 0.005

Dimensionless thickness Z

You

ng M

odul

us (G

Pa)

n=2

n=1

n=5

n=0.2

n=0.5

Fig. 2. Variation of Young's modulus through the dimensionless thickness Z of a P-FGM

plate

ACCEPTED MANUSCRIPT

20

-0.25

-0.23

-0.21

-0.19

-0.17

-0.15

-0.13

0 20 40 60 80 100

Power-law index n

Dim

ensi

onle

ss c

entre

def

lect

ion,

W

CPT[25], FSDT, TSDT

Fig. 3. Centre deflection versus power-law index n under load 025.0−=Q and aspect ratio

20=AR .

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21

0

100

200

300

400

500

600

2 4 6 8 10 12 14 16

Aspect ratio, AR

Inve

rse

of th

e di

men

sion

less

cen

tre d

efle

ctio

n, (a

/w)

CPT[25]

FSDT

TSDT

Fig. 4. Comparison of the centre deflection for different theories versus aspect ratio AR

under constant load 975.1 eq −= and power-law index 0=n .

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-0.5

-0.4

-0.3

-0.2

-0.1

0 20 40 60 80 100

Power-law index n

Dim

ensi

onle

ss c

entre

def

lect

ion,

W

CPT

FSDT

TSDT

Fig. 5. Centre deflection versus power-law index n under load 5.2−=Q and aspect ratio

2=AR .

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AR=20

-30

-20

-10

0

10

20

30

40

50

60

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Dimensionless thickness coordinate, Z

Dim

ensi

onle

ss a

xial

stre

sses

, �CPT

FSDT

TSDT

AR=4

-30

-20

-10

0

10

20

30

40

50

60

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT

AR=2

-30

-20

-10

0

10

20

30

40

50

60

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT

Fig. 6. Through the thickness axial stress σ at the center of the plate under load

400−=Q and power-law index 0=n .

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24

AR=20

-15

-5

5

15

25

35

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT

AR=4

-15

-5

5

15

25

35

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

stre

sses

, �

CPT

FSDT

TSDT

AR=2

-15

-5

5

15

25

35

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT



ACCEPTED MANUSCRIPT

25

AR=20

-15

-5

5

15

25

35

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT

AR=4

-10

0

10

20

30

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT

AR=2

-10

0

10

20

30

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT



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AR=20

-20

-10

0

10

20

30

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT

AR=4

-20

-10

0

10

20

30

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

stre

sses

, �

CPT

FSDT

TSDT

AR=2

-20

-10

0

10

20

30

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT



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27

AR=20

-20

-10

0

10

20

30

40

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

stre

sses

, �

CPT

FSDT

TSDT

AR=4

-25

-15

-5

5

15

25

35

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT

AR=2

-25

-15

-5

5

15

25

35

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

stre

sses

, �

CPT

FSDT

TSDT


400−=Q and power-law index 5.0=n .

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28

AR=20

-30

-20

-10

0

10

20

30

40

50

60

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT

AR=4

-30

-20

-10

0

10

20

30

40

50

60

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT

AR=2

-30

-20

-10

0

10

20

30

40

50

60

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5


Dim

ensi

onle

ss a

xial

str

esse

s, �

CPT

FSDT

TSDT


400−=Q and power-law index 01.0=n .

FGMnol

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