Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

51, 2, pp. 339-348, Warsaw 2013

NON-LINEAR ANALYSIS OF FUNCTIONALLY GRADED PLATES IN

CYLINDRICAL BENDING BASED ON A NEW REFINED SHEAR

DEFORMATION THEORY

Mokhtar Derras
Faculté de Technology, Département de Génie Mécanique, Algérie

Abdelhakim Kaci, Kada Draiche
Université de Sidi Bel Abbes, Laboratoire des Matériaux et Hydrologie, Algérie

Abdelouahed Tounsi
Université de Sidi Bel Abbes, Laboratoire des Matériaux et Hydrologie, Algérie, and

Faculté de Technology, Département de Génie Civil, Algérie

e-mail: tou abdel@yahoo.com

A new refined shear deformation theory for the nonlinear cylindrical bending behavior of
functionally graded (FG) plates is developed in this paper. This new theory is based on
the assumption that the transverse displacements consist of bending and shear components,
in which the bending components do not contribute toward shear forces and, likewise, the
shear components do not contribute toward bending moments. The theory accounts for a
quadratic variation of the transverse shear strains across the thickness, and satisfies the zero
traction boundary conditions on the top and bottom surfaces of the plate without using
shear correction factors. The plates are subjected to pressure loading, and their geometric
nonlinearity is introduced in the strain-displacement equations based on Von-Karman as-
sumptions. The material properties of plate are assumed to vary according to the power
law distribution of the volume fraction of the constituents. The solutions are achieved by
minimizing the total potential energy and the results are compared to the classical, the
first-order and other higher-order theories reported in the literature. It can be concluded
that the proposed theory is accurate and simple in solving the nonlinear cylindrical bending
behavior of functionally graded plates.

Key words: functional composites, plate, large deformation, energymethod

1. Introduction

Composite materials have been successfully used in aircraft and other engineering applications
formanyyears because of their excellent strength toweight and stiffness toweight ratios. Recen-
tly, advanced composite materials known as functionally graded material have attracted much
attention in many engineering applications due to their advantages of being able to resist high
temperature gradient while maintaining structural integrity (Koizumi, 1997). The functionally
graded materials (FGMs) are microscopically inhomogeneous, in which the mechanical proper-
ties vary smoothly and continuously from one surface to the other. They are usuallymade from
amixture of ceramics andmetals to attain the significant requirement of material properties.

Cylindrical bendinganalysis andalsonon-linear analysis ofFGplates are rare in literature re-
views.Qian et al. (2004) analysed plane strain thermoelastic deformations of a simply-supported
FGplate by ameshless local Petrov-Galerkinmethod. Bian et al. (2005) extended a recently de-
veloped plate theory using the concept of shape function of the transverse coordinate parameter
to investigate the cylindrical bending behavior of FG plates. Using the finite-element method,
Praveen and Reddy (1998) investigated static and dynamic responses of FG plates accounting



340 M. Derras et al.

for transverse shear strains, rotary inertia, andmoderately large rotations. Based on the higher-
order shear deformation plate theory, Reddy (2000) developedNavier’s solutions for rectangular
plates and finite-element models to study the non-linear dynamic response of FG plates. An
analytical solution was obtained by Woo and Meguid (2001) for large deflections of FG plates
and shallow shells under transverse mechanical loads and a temperature field. Using Reddy’s
higher-order shear deformation plate theory, Shen (2002) studied the non-linear bending of a
simply-supported FG rectangular plate subjected to mechanical and thermal loads. Yang and
Shen (2003a,b) investigated large deflection and postbuckling responses of FG rectangular pla-
tes under transverse and in-plane loads, and also they investigated non-linear bending of shear
deformable FG plates with various edge supports subjected to thermo-mechanical loads using
a semi-analytical approach. More recently, Shen (2007) presented nonlinear thermal bending
analysis for a simply-supported FG plate without or with piezoelectric actuators subjected to
the combined action of thermal and electrical loads using a two step perturbation technique.
We note that the majority of the above mentioned studies are concerned with analysis of func-
tionally graded plates wherematerial properties vary continuously through the plate thickness.
However, in recent studies, Jędrysiak and Michalak (2011) and Michalak and Wirowski (2012)
investigated the stability and dynamic behavior of thin plates with effective properties varying
in the midplane of the plates. The formulation of mathematical models of these plates is based
on a tolerance averaging technique (Woźniak et al., 2008).

The purpose of this study is to develop a new shear deformation plate theory for the non-
linear cylindrical bending behavior of FG plates which is simple to use. This theory is based
on the assumption that the in-plane and transverse displacements consist of bending and shear
components in which the bending components do not contribute toward shear forces and, like-
wise, the shear components do not contribute toward bending moments. The most interesting
feature of this theory is that it accounts for a quadratic variation of the transverse shear strains
across the thickness and satisfies the zero traction boundary conditions on the top and bottom
surfaces of the plate without using shear correction factors. Young’s modulus andmass density
of theFGplate are assumed to vary according to a power law distribution of the volume fraction
of the constituents whereas Poisson’s ratio is constant. The solution of the nonlinear cylindrical
bending problem is obtained via minimization of the total potential energy. To illustrate the
accuracy of the present theory, the obtained results are compared with those of the classical,
first-order and other higher-order theories reported in the literature.

2. higher order displacement theory

The displacements of amaterial point located at (x,y,z) in the plate under cylindrical bending
may be written as

u=u0(x,y)−z
prtw0
∂x
+Ψ(z)θx w=w0(x,y) (2.1)

where, u and w are displacements in the x, z directions, u0 and w0 aremidplanedisplacements,
θx rotation of the yz plane due to bending. Ψ(z) represents the shape function determining
the distribution of transverse shear strains and stresses along the thickness. The displacement
field of the classical thin plate theory (CPT) is obtained easily by setting Ψ(z) = 0. The
displacement of thefirst shear deformation theory (FSDT) is obtainedby setting Ψ(z)= z. Also,
the displacement of parabolic shear deformation theory (PSDT) by Reddy (1984) is obtained
by setting

Ψ(z)= z
(

1−
4z2

3h2

)

(2.2)



Non-linear analysis of functionally graded plates in cylindrical... 341

The sinusoidal shear deformation theory (SSDT) of Touratier (1991) is obtained by setting

Ψ(z)=
h

π
sin
πz

h
(2.3)

3. Present new refined shear deformation theory

This theory is based on the assumption that the in-plane and transverse displacements consist
of bending and shear components in which the bending components do not contribute toward
shear forces and, likewise, the shear componentsdonot contribute towardbendingmoments.The
theory presented is variationally consistent, does not require shear correction factor, and gives
rise to transverse shear stress variation such that the transverse shear stresses vary parabolically
across the thickness satisfying the shear stress free surface conditions.

3.1. Basic assumptions

The assumptions of the present theory for the nonlinear cylindrical bending of FG plate are
as follows:

(i) The displacements are small in comparisonwith the plate thickness and, therefore, strains
involved are infinitesimal.

(ii) The transverse displacement w includes two components of bending wb, and shear ws.
These components are functions of the coordinate x only

w(x,z) =wb(x)+ws(x) (3.1)

(iii) The transverse normal stress σz is negligible in comparison with the in-plane stresses σx.

(iv) The displacement u in the x-direction consists of extension, bending, and shears compo-
nents

u=u0+ub+us (3.2)

Thebendingcomponent ub is assumed tobe similar to thedisplacementgivenby theclassical
plate theory. Therefore, the expression for ub may be given as

ub =−z
∂wb
∂x

(3.3)

The shear component us gives rise, in conjunction with ws, to the parabolic variations of the
shear strain γxz and hence to the shear stress τxz through the thickness of the plate in such a
way that the shear stress τxz is zero at the top and bottom faces of the plate. Consequently, the
expression for us may be given as

us =−f(z)
∂ws
∂x

f(z)=−z
[1

4
−
5

3

(z

h

)2]

(3.4)

3.2. Kinematics

Based on the assumptions made in the preceding section, the displacement field can be
obtained using Eqs. (2.1)-(2.3) and (3.1) as

u(x,y,z) =u0(x,y)−z
∂wb
∂x
−f(z)

∂ws
∂x

w(x,y,z) =wb(x,y)+ws(x,y) (3.5)



342 M. Derras et al.

In the present study, we wish to investigate the effect of geometric non-linearity on the re-
sponse quantities. Therefore, the Von-Karman-type of geometric non-linearity is taken into
consideration in the strain-displacement relations. Substituting Eqs. (3.5) in the appropriate
strain-displacement relations (Fung, 1965) results in

εx = ε
0
x+zk

b
x+f(z)k

s
x γxz = g(z)γ

s
xz εz =0 (3.6)

where

ε0x =
∂u0
∂x
+
1

2

(∂wb
∂x
+
∂ws
∂x

)2

kbx =−
∂2wb
∂x2

ksx =−
∂2ws
∂x2

γsxz =
∂ws
∂x

g(z) =
5

4
−5
(z

h

)2
(3.7)

3.3. Constitutive relations

Consider a functionally graded plate which is made from amixture of ceramics andmetals.
It is assumed that the composition properties of FGM vary through the thickness of the plate.
The variation of material properties can be expressed as

P(z) =Pb+(Pt−Pb)Vt (3.8)

where P denotes a generic material property like modulus, and Pt and Pb denote the corre-
sponding properties of the top and bottom faces of the plate, respectively. Also Vt in Eq. (3.8)
denotes the volume fraction of the top face constituent and follows a simple power-law as

Vt =
(z

h
+
1

2

)n
(3.9)

where n (0 ¬ n ¬ ∞) is a parameter that dictates the material variation profile through the
thickness. Here we assume that the moduli E and G vary according to Eq. (3.8) and Poisson’s
ratio ν is assumed to be a constant.

The linear constitutive relations are

σx =Q11εx τxz =Q55γxz (3.10)

where (σx,τyx) and (εx,γxz) are the stress and strain components, respectively. Using the ma-
terial properties defined in Eq. (3.8), the stiffness coefficients Qij, may be expressed as

Q11 =
E(z)

1−ν2
Q55 =

E(z)

2(1+ν)
= g(z) (3.11)

The stress andmoment resultants of the FGM plate can be obtained by integrating Eq. (3.10)
over the thickness, and are written as

(Nx,M
b
x,M

s
x)=

h/2
∫

−h/2

σx
(

1,z,f(z)
)

dz Ssxz =

h/2
∫

−h/2

τxzg(z) dz (3.12)

Substituting Eq. (3.10) into Eq. (3.12) and integrating through the thickness of the plate, the
stress resultants are given as

Nx =A11ε
0
x+B11k

b
x+B

s
11k
s
x M

b
x =A11ε

0
x+D11k

b
x+D

s
11k
s
x

Msx =B
s
11ε
0
x+D

s
11k
b
x+H

s
11k
s
x S

s
xz =A

s
55γxz

(3.13)



Non-linear analysis of functionally graded plates in cylindrical... 343

where Aij,Bij, etc., are the plate stiffnesses defined by

{A11,B11,D11,E11,F11,H11}=
h/2
∫

−h/2

{1,z,z2,z3,z4,z6}Q11 dz

Bs11 =−
1

4
B11+

5

3h2
E11 D

s
11 =−

1

4
D11+

5

3h2
F11

Hs11 =
1

16
D11−

5

6h2
F11+

25

9h4
H11 {A55,D55,F55}=

h/2
∫

−h/2

{1,z2,z4}Q55 dz

As55 =
25

16
A55−

25

2h2
D55+

25

h4
F55

(3.14)

3.4. Solution procedure

The total potential energy Π of the FG plate is determined by the summation of strain
energy and the change in potential energy of the uniform externally applied pressure, and is
written as

Π =U+V (3.15)

Here, V is the potential energy of a plate under uniform pressure, and is equal to

V =

a
∫

−a

q(wb+ws) dx (3.16)

where q is the uniformly distributed load and 2a is the length of the FG plate. Also, the strain
energy U is defined as

U =
1

2

a
∫

−a

h/2
∫

−h/2

(σxεx+ τxzγxz) dxdz (3.17)

Now considering the boundary condition for the simply supported FG plate, the principal of
minimum potential energy is applied assuming a first guess solution for the considered displa-
cements (i.e. u0, wb and ws) over the mid-surface of the plate as

wb =ws =u0 =0 at x=−a,a

Mbx =M
s
x =0 at x=−a,a

(3.18)

The required mentioned displacement and rotation fields, which satisfy the simply supported
boundary conditions, are defined as











u0
wb
ws











=



























C sin
πx

a

Wb0 cos
πx

2a

Ws0 cos
πx

2a



























(3.19)

where C, Wb0, and W
s
0 are arbitrary parameters and are determined by minimizing the total

potential energy as

∂Π

∂(C,Wb0 ,W
s
0)
= 0 (3.20)

Equations (3.18) provides a set of three nonlinear equilibrium equations in terms of C, Wb0,
and Ws0 which should be solved. The obtained constants are then used to calculate the di-
splacements (Eq. (3.19)) and, subsequently, the strain and stresses are found using Eqs. (3.6)
and (3.10).



344 M. Derras et al.

4. Numerical results and discussion

To examine the nonlinear bending behavior of an structure, a plate consisting of aluminum and
alumina as the respective metal and ceramic substances of the FGM considered as an example.
Young’s modulus for aluminum is 70GPa while for alumina it is 380GPa. Note that Poisson’s
ratio is selected constant and equal to 0.3 for both constituents. For all analyses, the lower
surface of the plate is assumed to be rich inmetal (aluminum) and the upper surface to be rich
in ceramic (alumina).
The results obtained from the analysis are presented in dimensionless parametric terms of

deflections and stresses as follows:

• central deflection W =w/h

• loading parameter Q= qL4/(Ebh
4)

• axial stress σ=σxL
2/(Ebh

2)

• shear stress ∗τ = τxzL
2/(Ebh

2)

• thickness coordinate Z = z/h

where Eb is Young’s modulus of the metal (bottom surface) used in the functionally graded
material, and q is uniformly distributed pressure load. The analysis is performed on a FG plate
of length L = 2a = 200mm and thickness h = 10mm with the simply supported boundary
condition. The shear correction factor of FSDT is fixed to be K =5/6.
Figure 1 shows the inverse of the dimensionless deflection (L/w) of the homogeneous plate

(n = 0) as a function of the plate aspect ratio (L/h). It can be observed that the results
obtained by the present new refined shear deformation theory are identical to those of the first
shear deformation theory (FSDT), parabolic shear deformation theory (PSDT) and sinusoidal
shear deformation theory (SSDT). It is observed that theCPTmay be sufficient for thin plates.

Fig. 1. Comparison of the center deflection for different theories versus aspect ratio (L,h) under
constant load q=−0.56GPa and power-law index n=0

Figure 2 shows the dimensionless deflection of the center of the plate with aspect ratio of
(L/h = 20). As it is seen, the results obtained by the present new refined shear deformation
theory, FSDT, PSDT and SSDT coincide with the results of CPT.This is well explained by the
large plate aspect ratio (L/h=20). 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.
Figure 3 shows the dimensionless deflection of the center of the plate with aspect ratio of

(L/h = 2) using different plate 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, based on the figure, the CPT predicts lower deflection than the other theories,
and the predictions by shear deformation theories are close, though the latter gives the largest
deflection for the plate.



Non-linear analysis of functionally graded plates in cylindrical... 345

Fig. 2. Center deflection versus power-law index n under load Q=−20 and aspect ratio L/h=20

Fig. 3. Center deflection versus power-law index n under load Q=−20 and aspect ratio L/h=2

Fig. 4. Through-the-thickness axial stress σ at the center of the FG plate under load Q=−20 and
power-law index n=2: (a) L/h=20, (b) L/h=4, (c) L/h=2



346 M. Derras et al.

The results of the throrrasugh-the-thickness axial stress of theFGplate (n=2) are shown in
Fig. 4. It is observed that the results computed using the new refined shear deformation theory
are in a good agreement with those obtained by using FSDT, PSDT and SSDT. The results
obtained by all models (CPT, FSDT, PSDT, SSDT and the present) are in excellent agreement
even though the plate is very thin.
Figure 5 depicts through-the-thickness distributions of the shear stresses in the FG plate

(n = 2). It is observed from these figures that the through-the-thickness distributions of the
shear stresses are not parabolic. It is to be noted that the maximum value occurs at ,Z ∼=0.2,
not at the plate center as in the homogeneous case. The results obtained by all models (PSDT,
SSDT and the present) are in good agreement, particularly with PSDT.

Fig. 5. Through-the-thickness shear stress τ at the center of the FG plate under load Q=−20 and
power-law index n=2: (a) L/h=20, (b) L/h=4, (c) L/h=2

Fig. 6. Dimensionless centre deflection versus uniform pressure load with L/h=10

Figure 6 illustrates variation of the non-dimensional center deflection of the FG square plate
with different values of the power-law index n subjected to a uniform transverse load. The
results show that a pure metal plate has the highest deflection. This is expected because the
fully metallic plate is one with lower stiffness than ceramic and FG plates.



Non-linear analysis of functionally graded plates in cylindrical... 347

5. Conclusion

The nonlinear cylindrical bending of FG plates under pressure loading was studied using a
new refined shear deformation theory. Fundamental equations for thick square FG plates have
been obtained using the Von-Karman theory for large deflections. The solution of the nonlinear
cylindrical bending problem is obtained via minimization of the total potential energy. The
accuracy and efficiency of the present theory have been demonstrated for nonlinear bending
analysis of simply supported FG plates. It can be concluded that the present theory is not only
accurate but also efficient in predicting the nonlinear cylindrical bending response of FG plates
compared to other shear deformation plate theories such as FSDT, PSDT and SSDT.

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348 M. Derras et al.

Nieliniowa analiza zginania walcowego płyt wykonanych z materiału gradientowego

w oparciu o nową teorię odkształcenia postaciowego

Streszczenie

W niniejszym artykule zaprezentowana została nowa, udoskonalona teoria odkształcenia postaciowe-
go dla przypadku zginania walcowego płyt wykonanych z funkcjonalnie gradientowego (FG) materiału.
Teorię tę oparto na założeniu, że przemieszczenie poprzeczne jest wynikiem efektu zginania i ścinania,
ale zginanie nie wpływa na siły tnące ani ścinanie nawartośćmomentu gnącego.Teoriawprowadza potę-
gową zależność czwartego stopnia odkształceń postaciowych w funkcji grubości płyty i spełnia warunek
brzegowyzerowanianaprężeń ścinającychna zewnętrznychpowierzchniachpłytybez użyciawspółczynni-
ków korekcyjnych.W pracy zbadano płyty poddane ciągłym obciążeniom zewnętrznym, a uwzględniona
nieliniowośćmiała charakter geometrycznywynikający z opisu związkuvonKarmanapomiędzy odkształ-
ceniem i przemieszczeniem. Gradientowe właściwości materiału ujęto potęgowym rozkładem poszczegól-
nych składnikówwzdłuż grubości płyty.Rozwiązania otrzymanopoprzezminimalizację całkowitej energii
potencjalnej przy zginaniu, a uzyskane wyniki porównano ze znanymi w literaturze rezultatami klasycz-
nej teorii odkształcenia pierwszego oraz wyższych rzędów. Przedstawiona analiza potwierdziła, że nowo
zaproponowana teoria jest dokładna i prostaw rozwiązywaniunieliniowegoproblemu zginaniawalcowego
płyt wykonanych z materiałów gradientowych.

Manuscript received February 20, 2012; accepted for print June 19, 2012