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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
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers
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1
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
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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ACCEPTED MANUSCRIPT
18
Fig. 1. The geometry of a typical FGM plate
ACCEPTED MANUSCRIPT
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 .
ACCEPTED MANUSCRIPT
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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22
-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 .
ACCEPTED MANUSCRIPT
23
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
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 .
ACCEPTED MANUSCRIPT
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
Dim
ensi
onle
ss a
xial
str
esse
s, �
CPT
FSDT
TSDT
Fig. 7. Through the thickness axial stress σ at the center of the plate under load
400−=Q and power-law index 100=n .
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
Dim
ensi
onle
ss a
xial
str
esse
s, �
CPT
FSDT
TSDT
Fig. 8. Through the thickness axial stress σ at the center of the plate under load
400−=Q and power-law index 2=n .
ACCEPTED MANUSCRIPT
26
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
Dim
ensi
onle
ss a
xial
str
esse
s, �
CPT
FSDT
TSDT
Fig. 9. Through the thickness axial stress σ at the center of the plate under load
400−=Q and power-law index 1=n .
ACCEPTED MANUSCRIPT
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
Dim
ensi
onle
ss a
xial
stre
sses
, �
CPT
FSDT
TSDT
Fig. 10. Through the thickness axial stress σ at the center of the plate under load
400−=Q and power-law index 5.0=n .
ACCEPTED MANUSCRIPT
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
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
Dimensionless thickness coordinate, Z
Dim
ensi
onle
ss a
xial
str
esse
s, �
CPT
FSDT
TSDT
Fig. 11. Through the thickness axial stress σ at the center of the plate under load
400−=Q and power-law index 01.0=n .