material and geometric nonlinear analysis … · josé s. moita, aurélio l. araújo, cristóvão...

10th International Conference on Composite Science and Technology

ICCST/10

A.L. Araújo, J.R. Correia, C.M. Mota Soares, et al. (Editors)

© IDMEC 2015

MATERIAL AND GEOMETRIC NONLINEAR ANALYSIS OF FUNCTIONALLY GRADED PLATE-SHELL TYPE STRUCTURES

José S. Moita† *, Aurélio L. Araújo †, Cristóvão M. Mota Soares †, Carlos A. Mota Soares †

*FANOR, Faculdades Nordeste, Av. Santos Dumont, 7800, Fortaleza, Brazil

[email protected]

†LAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa

Av. Rovisco Pais, 1049-001 Lisboa, Portugal [email protected]

This work is dedicated in honour of Professor J.N. Reddy on his 70th birthday and for his contribution and impact to

research and education on mechanics of advanced composite materials and structures. The authors also express their

gratitude for his friendship and scientific advices.

Key words: Functionally Graded Materials, Finite Element, Nonlinear Analysis.

Summary: A nonlinear formulation for general Functionally Graded Material (FGM) plate-

shell type structures is presented. The formulation accounts for geometric and material

nonlinear behaviour of these structures. Using the Newton-Raphson incremental-iterative

method, the incremental equilibrium path is obtained, and in case of snap-through

occurrence the automatic arc-length method is used. This simple and fast element model is a

non-conforming triangular flat plate-shell element with 24 degrees of freedom for the

generalized displacements. It is benchmarked in the solution of some illustrative plate- shell

examples and the results are presented and discussed with numerical alternative models.

1 INTRODUCTION

In an effort to develop the super heat resistant materials, Koizumi [1] first proposed the

concept of Functionally Graded Material (FGM). Typical FGM plate-shell type structures are

made of materials which are characterized by a continuous variation of the material

properties over the thickness direction by mixing two different materials, metal and ceramic.

The metal–ceramic FGM plates and shells are widely used in aircraft, space vehicles, reactor

vessels, and other engineering applications.

Structures made of composite materials have been widely used to satisfy high

performance demands. In such structures stress singularities may occur at the interface

between two different materials. In contrast, in FGM plate-shell structures the smooth and

continuous variation of the properties from one surface to the other eliminates abrupt changes

in the stress and displacement distributions.

In certain applications, structures can experience large elastic deformations and finite

rotations. Geometric nonlinearity plays a significant role in the behaviour of a plate or shell

structures, especially when it undergoes large deformations. Also material nonlinearity has a

significant role in the behaviour of these structures. However a review of the literature shows

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José S. Moita, Aurélio L. Araújo, Cristóvão M. Mota Soares,Carlos A. Mota Soares

2

that few studies have been carried out in nonlinear bending response of plates and shells. This

is the main reason to the present work, which include the behaviour response of this type of

structures due to the material and geometric nonlinearities.

Research in FGM structures has been done in the recent years. Among others, here we cite

the following works: Reddy and Chin [2] analyzed the dynamic thermo elastic response of

functionally graded cylinders and plates. Praveen and Reddy [3] carried out a nonlinear

thermoplastic analysis of functionally graded ceramic–metal plates using a finite element

model based on the First-Order Shear Deformation Theory (FSDT). Reddy [4] studied the

bending and vibration analyses of FGMs plates. Woo and Meguid [5] provided an analytical

solution for large deflection of FGM plates and shells under mechanical and thermal loading.

Yang and Shen [6, 7] carried out the analyses of nonlinear bending and post buckling

behavior for FGM plates under thermo mechanical load and with various boundary

conditions. Ma and Wang [8] investigated the nonlinear thermal bending and post-buckling

of circular functionally graded plates using FSDT and the Third -Order Shear Deformation

Theories (THSDT). Na and Kim [9], examined the effect of thermal loading and uniform

pressure on the bending response of FGM plates. Reddy and Arciniega [10, 11] presented the

thermo-mechanical buckling, as well as bending and free vibration analysis, of FGM plates.

The same authors [12] carried out the large deformation analysis of FGM shells. Kim et al.

[13] presented the geometric nonlinear analysis of functionally graded material plates and

shells using a four-node quasi-conforming shell element. Barbosa and Ferreira [14]

performed the geometric nonlinear analysis of functionally graded (FGM) plates and shells,

using the Marguerre shell element.

The plasticity formulation follows a composite model proposed by Tamura et al. [15]

referred as the TTO model. Nakamura et al. [16] and Gu et al. [17] proposed an inverse

analysis procedure based on the Kalman filter and instrumented micro-indentation. Jin et al.

[18] studied the fracture issues in elastic–plastic FGMs.

A comprehensive review of the various methods employed to study the static, dynamic

and stability behaviour of FGM plates was recently presented by Swaminathan et al. [19]

considering analytical and numerical methods.

In this paper, a large deformation analysis for functionally graded plate-shell structures is

presented. The formulation includes material and geometric nonlinearities and is

implemented in a finite element model based on a non-conforming triangular flat plate-shell

finite element in conjugation with the Reddy’s third-order shear deformation theory.

2 FORMULATION OF FGM MODEL

A FGM is made by mixing two distinct isotropic material phases, for example a ceramic

and a metal. The material properties of an FGM plate/shell structures are assumed to change

continuously throughout the thickness, according to the volume fraction of the constituent

materials. Power-law function [20] and exponential function [21] are commonly used to

describe the variations of material properties of FGM. However, in both power-law and

exponential functions, the stress concentrations appear in one of the interfaces in which the

material is continuously but rapidly changing. Therefore, Chung and Chi [22] proposed a

sigmoid FGM, which was composed of two power-law functions to define a new volume

fraction. Chi and Chung [23] indicated that the use of a sigmoid FGM can significantly

reduce the stress intensity factors of a cracked body.

To describe the volume fractions, the power-law function, and sigmoid function are used

here. Thus, in this work, two models for estimating the effective material properties of the


3

FGM at a point are considered.

2.1 Power-law function: P-FGM

The volume fraction of the ceramic phase is defined according to the power-law [20]:

p

ch

z5.0)z(V

(1)

being z(-h/2; h/2), h the thickness of structure, and the exponent p a parameter that defines

gradation of material properties across the thickness direction.

Thus, the metal volume fraction is

kcm V0.1)z(V (2)

2.2 Sigmoid function: S-FGM

The volume fraction uses two power-law functions which ensure smooth distribution of

stresses [22]:

p

c2h

z2h

2

10.1)z(V

for

2

hz0 (3a)

p

m2h

z2h

2

1)z(V

for 0z

2

h (3b)

In the present work, the continuous variation of the materials mixture is approximated by

the using a certain number of layers throughout the thickness direction (virtual layer

approach). In this sense, the previous equations can be written for each layer as follows (P-

FGM):

p kc

h

z5.0V

; k

ckm V0.1V (4)

where z is the thickness coordinate of mid-surface of each layer

Once the volume fraction kcV and

kmV have been defined, the material properties (H) of

each layer of a FGM can be determined by the rule of mixtures.

mkmc

kc

k H VH VH (5)

where H stands for Young’s modulus E, the Poisson’s ratio υ, the mass density ρ, or any

other mechanical property.

Figure 1 show the variation of Young’s modulus E through the thickness, obtained using

the sigmoid function and 20 layers.

3 NONLINEAR ANALYSIS THEORY

The objective of this work is to investigate the geometrically non-linear behaviour of


4

FGM plate-shell type structures under mechanical loading. The present theory considers large

displacements with small strains.

Figure 1. Variation of Young’s modulus through the thickness.

3.1 Displacement Field and Strains

The displacement field is based on the Reddy’s third-order shear deformation theory [24]

x

w y,xc zy,x zy,xuz,y,xu 0

y13

yc0

y

w y,x c zy,x zy,xvz,y,xv 0

x13

xc0 (6)

y,xwz,y,xw 0

where 0cc ,w, vu00

are displacements of a generic point in the middle plane of the core layer

referred to the local axes - x,y,z directions, ,, θθ yx are the rotations of the normal to the

middle plane, about the x axis (clockwise) and y axis (anticlockwise), xwyw 00 ,

are the slopes of the tangents of the deformed mid-surface in x,y directions, and 2

1 h 34c ,

with h denoting the total thickness of the structure.

To the geometrically nonlinear behaviour, the Green’s strain tensor is here considered. Its

components are conveniently represented in terms of the linear and nonlinear parts of the

strain tensor as

NLL (7)

The linear strain components associated with the displacement field defined above, can be

represented in a synthetic form as [25]:

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

7.00E+10 9.00E+10 1.10E+11 1.30E+11 1.50E+11

z/h

Young's modulus

p=0.2

p=0.5

p=1.0

p=2.0


5

*s

2s

b3

bmL

z

z z + (8)

or in de developed form, as:

2

2y

1

y0

x

w

xcz

x

z

x

u 03

xx

2

2x

13x0

y

w

ycz+

y

z

y

v = 0

yy

y x

w2

xy cz+

x

+

y

z+

x

v

y

u 02

xy

1xy00 3

xy+

x

wcz

x

w 0

y220

yxz

y

wcz

y

w 0

x220

xyz (9)

where2

2 h4c .

The nonlinear strain components are given by

222NL

x

w

x

v

x

u

2

1=

xx

222NL

y

w

y

v

y

u

2

1=

yy

y

w

x

w

y

v

x

v

y

u

x

u=NL

xy

z

w

x

w

z

v

x

v

z

u

x

u=NL

xz

z

w

y

w

z

v

y

v

z

u

y

u=NL

yz (10)

3.2 Constitutive Relations of FGM Structures

The stress-strain relations for each layer k, can be written as follows

kkk Q (11)


6

where Tyzxzxyyxk is the stress vector and T

yzxzxyyxk is the

strain vector, kQ is the elasticity matrix, whose non-zero elements are given by:

2k

k

2211

1

EQQ

11k

2112 QQQ

k

k

k

665544 G)1( 2

EQQQ

(12)

For each layer, linear elastic constitutive equation is given by [25]:

k

*s

s

*b

b

m

kss

ss

k

*

*

EC000

CA000

00GED

00ECB

00DBA

Q

Q

M

M

N

(13)

where for instance k4ijkij H QD , with 4zzH 4

k4

1k4

Equation (13) can be written in the following form

kkk Dˆ (14)

where k̂ are the resultant forces and moments.

4 ELASTO-PLASTIC FORMULATION

The yield condition can be written in the form

0f,F Y (15)

where the yield level, Y , can be a function of the hardening parameter κ.

From the Huber-Mises criterion, for the case of an isotropic material the effective stress

is given by

2232

132

1222112

222

112 3 3 3 f (16)

4.1 Stress-strain relations in elastoplasticity

The elasto-plastic increments of total strains can be calculated by summing up the elastic

and the plastic strain components,

pe d d d (17)

4.2 Flow rule

The plastic strain increment is defined as proportional to the stress gradient of a plastic

potential Q, which is taken equal to the yield surface condition for an associated flow rule:


7

d

dε d p Q

(18)

where is a proportional constant that determines the magnitude of the plastic strain

increment and ddQ defines its direction to be perpendicular to the yield surface.

Introducing the flow vector dda Q and p' ddHA where the hardening

parameter A can be taken from uniaxial testing, it can be obtained

a aA

d TQ

QaT

(19)

a QaA

dε a Qaε d

T

Tp

(20)

The incremental constitutive elastoplastic relation is given by, [26, 27]

dQdd(Qd eppe ) (21)

Introducing the equations (11) and (20) into equation (21), comes:

d

aQaA

Qa aQQ d

T

T

(22)

From this equation the elastoplastic constitutive matrix is obtained as:

a QaA

Qa a QQQ

T

Tep

(23)

EC000

CA000

0

0

0

kss

ss

0GED

0ECB

0DBA

D epepep

epepep

epepep

epk (24)

where for instance k

3epijk

epij HQC , with 3zzH 3

k3

1k3

The present work uses an extended Tamura–Tomota–Ozawa (TTO) model to describe the

elastic–plastic behaviour of ceramic/metal FGM. Ceramic materials are, in general, brittle

materials of relatively higher elastic modulus and strength than those of metallic materials,

which have typically ductile properties. Thus the ceramic constituent in FGM is assumed to

be elastic when deformation takes place. The elasto-plastic deformation occurs mainly by the

plastic flowing of the metallic constituent. It is adopted the modified rule-of-mixture which

uses the stress-strain transfer parameter q, which depends on the constituent material

properties and the microstructural interaction within the composite, and is given by [18]:

mc

mcq

(25)


8

Using this parameter, the thickness variation of Young’s modulus and the yield stress, ,Y

may be obtained as follows [18]:

c

c

mmcc

m

cmm

k VEq

EqVEV

Eq

EqEV E (26)

c

m

c

c

mmYm

kY V

E

E

Eq

EqV (27)

5 VIRTUAL WORK PRINCIPLE

The governing equations of the nonlinear problem are obtained from the principle of

virtual work in conjugation with an updated Lagrangian formulation [28, 29]. A reference

configuration is associated with time t and the actualized configuration is associated with the

current time 't t t . Thus we have:

tt

tt

ijttt

Vtij

ttt Vd S (28)

where the first and second member are the internal and external virtual work, respectively,

and ij and ijS are the Green-Lagrange strain components and the second Piola-Kirchhoff

stress components respectively.

Taking the Virtual Work Principle equation (28) and doing the incremental decomposition

of the stress and strain tensors, we obtain a new equation in an incremental form, and

linearized.

Vd e Vd Vd e e C tijt

V

ijttt

tt

ijt

V

ijtt

ijt rst

V

ijrstttt

(29)

where ij rsC is the constitutive tensor, ijt are the incremental components of the Cauchy

stress tensor, ijt e are the incremental components of the linear strain vector, and ijt are the

incremental components of the nonlinear strain vector.

5.1 Finite Element Formulation

In the present works it is used a non-conforming triangular plate/shell finite element

model having three nodes and eight degrees of freedom per node: the displacements

iii 000 , w , vu , the slopes i0i

xw , yw0 and the rotations ziyixi , θ, θθ . The

rotation zi is introduced to consider a fictitious stiffness coefficient ZK to eliminate the

problem of a singular stiffness matrix for general shape structures [27]. The element local

displacements, rotations and slopes, are expressed in terms of nodal variables through shape

functions iNj given in terms of area co-ordinates ,L i [27]. The displacement field can be

represented in matrix form as:

e

i( a NZdN Zu = ) =3

1=ii ;

ei a NdN d = =

3

1=ii (30)


9

where the appropriate matrix Z containing powers of zk, and the element and nodal

displacement vectors, ae and id respectively, are given in [25,30].

The membrane, bending and shear strains, as well the higher order bending and shear

strains can be represented by:

em

m aB ; eb

b aB ; es

s aB ; e*b

baB ;

e*a B *s

s (31)

where them

B , bB , *b

B , s

B , *s

B , are components of the strain-displacement matrix ,B and

are given explicitly in [30].

The Virtual Work Principle equation (29) applied to a finite element can be written in the

form:

N

1k eAt

kk

1kh

etLk

tLkt

eΔttt

N

1k eAt

kk

1kh eAt

kk

1kh

etLk

tNLkt

etLktk

Lkt

dAdz δ

dAdz δ dAdz Q δ

(32)

From this equation we obtain the element linear stiffness matrix ,eLK the element

geometric stiffness matrix eK , as well as element external load vector

eextF and the element

internal force vector ,eintF and their definition are explicitly given in [30]. For the element

geometric stiffness matrix ,eK

et A

Te etdA GGK (33)

the matrices G and τ are obtained with a fully development, in the same way as shown in [31]

for the case of eight node isoparametric finite element.

These matrices and vectors are initially computed in the local coordinate system attached

to the element. To solve general structures, local - global transformations are needed [27].

Doing these transformations, and after adding the contributions of all the elements in the

domain, the system equilibrium equations are obtained as:

1iint

tt

ttexttti1i

L

tt

tt

FFqKK (34)

Using the Newton-Raphson incremental-iterative method [28,29], the incremental

equilibrium path is obtained, and in case of snap-through occurrence the automatic arc-length

method is used [25,32]. Eq. (34) is then written in the form:

1iint

tt

tt

0ext

i1ii1iL

tt

tt

FF qKK (35)

and an additional equation is employed to constrain the length of a load step:

2iTi l qq (36)


10

where 0extF is a fixed (reference) load vector, is a load factor, is the incremental load

factor within the load step, and l is the arc-length.

6 APPLICATIONS

6.1 Simply Supported Square S- FGM Plate.Linear Analysis

A simply supported square (axa) plate under uniformly distributed load of intensity 2

0 m/N 0.1q , presented by Kim et al. [13], is analyzed using the present model. The

mechanical properties of the isotropic ceramic material are Ec = 151 x 109 N/m

2, cν = 0.3 and

for isotropic metal material are Em = 70 x109 N/m

2, mν = 0.30. The side to thickness ratio is

a/h=100. The S-FGM plate is modelled by using different finite element meshes from 4x4

(32 triangular elements) to 24x24 (1152 triangular elements). In this application and all the

the FGM cases, the layered approach was carried out with 10 layers. This value for the

number of layers has been chosen based on a convergence study.

The non-dimensional centre deflections w*= 40

3c a qh E w are shown in Table 1. From

this table it can be observed that the results given by the present finite element FGM model

using a 24x24 element mesh have an excellent agreement with the ones obtained by a FSDT

Navier analytical solution [13].

For the same application, Figure 2 shows the centre deflection as a function of the power-

law index. The discrepancies between the results of present model and the analytical solution

are about of 0.4%. Also is observed that under the same load, the case of p=10 has the largest

centre deflection. This is due to the fact that it has the lowest Young’s modulus.

Mesh Normalized deflection Discrepancy [%]

8x8 0.06209 7.6

12x12 0.06572 2.2

16x16 0.06660 0.9

20x20 0.06680 0.6

24x24 0.06699 0.3

Analytical 0.06721

Table 1. Normalized centre deflection of S-FGM plate (index p=10).

6.2 Geometrically nonlinear analysis of a functionally graded plate strip

A cantilevered P-FGM plate strip subjected to a bending distributed moment on the other

end is considered. The material properties for the ceramic and metal constituents are Ec = 151

x 109 N/m

2, cν = 0.3, Em = 70 x10

9 N/m

2, mν = 0.30. The geometry of the plate is L =12.0

m, b = 1.0 m, h = 0.1 m and the total bending moment MRef = 65, 886.17926 N.m. The strip

is modelled with a 32x4 finite element mesh with 256 triangular elements, with ten layers,

having a total of 642 free Degrees OF Freedom (DOF). Figure 3 show the results for the tip

deflection due to a bending distributed moment M for the different power-law p-index

considered, obtained by the present model (PM). In the same figure are presented discrete

results taken from Arciniega and Reddy [12] with a 1x8Q25 element mesh with a total of


11

1120 free DOF. It is observed a good agreement between both solutions. In Figure 4 are

shown the initial and deformed shapes of the plate strip for specific load levels, in the case of

p-index p=2.0

Figure 2. Centre deflection versus power law index.

Figure 3. Tip-deflection W vs. end moment M for the FGM plate strip.

6.3 Geometrically nonlinear analysis of a hinged functionally graded cylindrical panel

A S-FGM cylindrical shell panel represented in Figure 5 has the straight sides simply-

supported (hinged) and the curved sides free. The geometry is defined by: R= 2.54 m, L =

0.508 m and subtended angle 2 =0.2 rad, h=0.0126 m, and it is modelled by a 16x16 finite

element mesh, considering ten layers. The constituents are zirconia and aluminium and

several power-law p-indexes are considered.

Figure 6 shows the centre deflection due to a centre point applied load P for the different

p-index considered, and as expected, the case of pure metal has the largest centre deflection.

This is due to the fact that it has the lowest Young’s modulus. Also, when the load-

displacements curves obtained with the present model are compared with those obtained of

Kim et al. [13], a very good agreement is observed. Here we present the results comparison

0 1 2 3 4 5 6 7 8 9 10

0.06

0.061

0.062

0.063

0.064

0.065

0.066

0.067

0.068

S-FGM power law index No

rmal

ized

cen

ter

def

lect

ion

Analytical solution [13]

Present model

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10 12 14 16

M /

Mre

f

Deflection at the tip

CERAMIC (P M)

CERAMIC Ref. [12]

p=0.2 (PM)

p=0.2

p=0.5 (PM)

p=0.5

p=1.0 (PM)

p=1.0

p=2.0 (PM)

p=2.0

p=5.0 (PM)

p=5.0

METAL (PM)

METAL


12

for ceramic and p-index=10.0 cases, Figures 6a and 6b, respectively.

Figure 4. Initial and deformed shapes of plate strip.

Figure 5. Cylindrical panel.

0

2

4

6

8

10

12

-8 -3 2 7 12

M/Mref=0.0

M/Mref=0.25

M/Mref=0.40

M/Mref=0.50

0

20000

40000

60000

80000

100000

120000

140000

-5 5 15 25 35

Cen

tral

po

int

load

(N

)

Central deflection Wc (mm)

Metal p=0.2

p=2.0 p=10.0

Ceramic


13

Figure 6. Load-deflection curve for different power law index.

Figure 6a. Load-deflection curve for ceramic.

Figure 6b. Load-deflection curve for power law index p=10.0

6.4 Nonlinear analysis of a simply-supported square plate

The linear and nonlinear bending behaviour of a square (axa) plate with the four sides

simply supported is analyzed using the present model. This application is proposed by

Supriyono and Aliabadi [33] and presented by Waidemam [34]. The mechanical properties of

the isotropic material are E = 100 x 106 N/m

2, ν = 0.316 and uniaxial yield stress

0.12500Y N/m2. The side is a=1.0 m, and the thickness is h=0.01 m. The plate is

modelled by using a 16x16 finite element mesh, corresponding to 512 triangular finite

elements. Figure 7 shows the centre deflection w*=w/h due to a uniformly distributed

pressure load44 h E/a q*q .

0

20000

40000

60000

80000

100000

120000

140000

0 5 10 15 20 25 30

Cen

ter

po

int

load

(N

)

Center deflection Wc (mm)

Ceramic Present model

Kim et al. [13]

0

20000

40000

60000

80000

100000

120000

140000

0 10 20 30 40

Cen

ter

poin

t lo

ad (

N)


p=10.0

Present model

Kim et al. [13]


14

It can be observed very good correlations between solutions obtained with the present

model (PM) and the model based in boundary elements used by Waidemam [34], for Linear,

Geometrically Nonlinear (G_NLinear), Material Nonlinear (M_NLinear) and Material and

Geometrically Nonlinear (G+M_NLinear) analyses.

Following the conclusions taken from Figure 7, two similar applications are extended for a

FGM structure, and proposed for benchmark purposes in sections 6.5 and 6.6.

Figure 7. Centre point deflection for different analyses.

6.5 Simply Supported Square P- FGM Plate

A simply supported square (axa) with side dimension a=1.0 m and side to thickness ratio

a/h=50, is under a uniformly distributed load of intensity 2

0 m/N 1000q . The material

properties for the ceramic and metal constituents are Ec = 350 GPa, cν = 0.24, Em = 70 GPa,

mν = 0.30. The yield stress for pure metal is MPa 225Y , the ratio q/Ec is taken as 0.80,

based on the micro- indentation experiments by Gu et al. [17], and is considered q=290 GPa.

This P-FGM plate with p-index power-law function p=1.0, is analyzed considering four

different material behaviours: elastic deformation, materially nonlinear deformation,

geometrically nonlinear deformation and geometrically and material nonlinear deformation.

In Figure 8 the results obtained with the present model are shown, considering ten layers.

6.6 Geometrically nonlinear analysis of a clamped functionally graded cylindrical panel

A square clamped P-FGM cylindrical shell panel with constituents silicon nitride and

stainless steel, side-to-thickness ratio a/h=100, side-to-radius ratio a/R=0.1, and several

power law exponents p are presented. The material properties for the ceramic and metal

constituents are Ec = 322.27 GPa, cν = 0.24, Em = 207.78 GPa, mν = 0.3177. The panel is

modelled by a 12x12 finite element mesh (288 triangular elements), and considering ten

layers. Using the present model, Figure 9 shows the centre deflection due to a centre point

applied load P for the different power-law p-index considered, and as expected, the case of

0 0.5 1 1.5 2

0

2.5

5

7.5

10

12.5

15

17.5

20

W*

q*

Linear (PM) Linear [34 ]

G_NLinear (PM) G_NLinear [34 ]

M_NLinear (PM) M_NLinear [34 ]

G+M_NLinear (PM) G+M_NLinear [34 ]


15

pure metal has the largest centre deflection, and the case of pure ceramic has the smallest

centre deflection. Also is observed that the deflection curves of the functionally graded

material panels are located between those of the metal and ceramic panels.

Next, the same square clamped P-FGM cylindrical panel, with yield stress for pure metal

MPa 250Y , and considering q=280 GPa, and p-index p=1.0, is analyzed considering

four different material behaviours: linear elastic, material nonlinear, geometrically nonlinear

and geometric and materially nonlinear deformations. In Figure 10 the solutions given by the

present model are shown.

Figure 8. Centre point deflection of a square P-FGM plate for different analyses.

Figure 9. Load-deflection curves of a P-FGM clamped cylindrical panel

for different power law index.

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 20 40 60 80

pre

ssure

load

(kP

a)

Wc (mm)

Linear

M_NLinear

G_NLinear

G + M_NLinear

0

50000

100000

150000

200000

250000

300000

350000

400000

450000

500000

0 10 20 30 40 50

Cen

ter

po

int

load

(N

)


Ceramic

p=0.2

p=2.0

p=10.0

Steel


16

Figure 10. Centre point deflection of a P-FGM clamped cylindrical panel for different analyses.

7 CONCLUSIONS

A finite element model for the static analysis of functionally graded material plate-shell

structures is extended for the material and geometrically nonlinear analysis. The model is

based on the Reddy’s third order shear deformation theory, implemented in a non-

conforming flat triangular plate/shell element with 24 degrees of freedom for the generalized

displacements. The continuous variation of the mechanical properties through the thickness is

approximated by only ten layers for all FGM applications, which preserves the reduced

computational time of the model.

From the results obtained by the present model, a very good accuracy is found with the

available comparing solutions obtained by alternative models. Two examples are also

proposed which can be used as benchmark tests for FGM plate-shell structures with material

and geometrically nonlinear behaviour.

ACKNOWLEDGEMENTS: This work was supported by FCT, Fundação para a Ciência e

Tecnologia, through IDMEC, under LAETA, project UID/EMS/50022/2013.

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http://www.teses.usp.br/teses/disponiveis/18/18134/tde-10092008-102729/en.php

http://www.teses.usp.br/teses/disponiveis/18/18134/tde-10092008-102729/en.php

material and geometric nonlinear analysis … · josé s. moita, aurélio l. araújo, cristóvão...

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