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2010 16: 2147 originally published online 17 June 2010Journal of Vibration and ControlHesham Hamed Ibrahim and Mohammad Tawfik

Aerodynamic and Thermal LoadsLimit-cycle Oscillations of Functionally Graded Material Plates Subject to

  

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Limit-cycle Oscillations of Functionally GradedMaterial Plates Subject to Aerodynamic and ThermalLoads

HESHAM HAMED IBRAHIMSpace Division, National Authority for Remote Sensing and Space Sciences, Cairo, Egypt, 11769

MOHAMMAD TAWFIKDepartment of Aerospace Engineering, Cairo University, Giza, Egypt(mohammad.tawfik@gmail.com)

(Received 7 June 2007� accepted 15 June 2009)

Abstract: The nonlinear flutter and thermal buckling of a functionally graded material (FGM) plate panelsubjected to combined thermal and aerodynamic loads are investigated using a finite element model based onthe thin plate theory and von Karman strain-displacement relations to account for moderately large deflection.The thermal load is assumed to be steady-state constant temperature distribution, and the aerodynamic pres-sure is modeled using the quasi-steady first-order piston theory. The governing nonlinear equations of motionare obtained using the principle of virtual work adopting an approach based on the thermal strain being a cu-mulative physical quantity to account for temperature dependent material properties. The static nonlinearequations are solved by Newton-Raphson numerical technique to get the thermal post-buckling deflection.The dynamic nonlinear equations of motion are transformed to modal coordinates to reduce the computa-tional efforts. The Newmark implicit integration scheme is employed to solve the second order ordinarydifferential equations of motion. Finally, the buckling temperature, post-buckling deflection and the nonlin-ear limit-cycle oscillations of an FGM panel are presented, illustrating the effect of volume fraction exponent,dynamic pressure, temperature rise, and boundary conditions on the panel response.

Key words: Functionally graded materials, nonlinear flutter, thermal buckling

1. INTRODUCTION

The external skin of high speed flight vehicles experiences high temperature rise due to aero-dynamic heating, which can induce thermal buckling and dynamic instability. In general,thermal buckling does not indicate structural failure. However, the thermal large deflectionof the skin panels can change its aerodynamic shape causing reduction in the flight perfor-mance.

A comprehensive literature review on thermally induced flexure, buckling, and vibra-tion of plates and shells is presented by Tauchart (1991) and Thornton (1993). Gray and

Journal of Vibration and Control, 16(14): 2147–2166, 2010 DOI: 10.1177/1077546309353366

��2010 SAGE Publications Los Angeles, London, New Delhi, Singapore

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2148 H. H. IBRAHIM and M. TAWFIK

Mei (1991) investigated the thermal post-buckling behavior and free vibration of thermallybuckled composite plates using the finite element method. Shi and Mei (1999) solved theproblem of thermal post-buckling of composite plates with initial imperfections using thefinite element modal method. Jones et al. (1980) investigated the linear and nonlinear dy-namic behavior of plates at elevated temperatures. They presented analytical solutions for thethermal buckling and post-buckling behavior of a plate strip. A general formula is also pre-sented that links the fundamental frequency of vibration to the critical buckling temperatureand the corresponding frequency of the unheated plate. Shi et al. (1999) investigated the ther-mal post-buckling behavior of symmetrically laminated and anti-symmetric angle-ply plates,and the deflection of asymmetrically laminated composite plates under combined mechani-cal and thermal loads. A finite element formulation in modal coordinates was developed forthe nonlinear thermal post-buckling of thin composite plates. The quantative contribution ofeach linear buckling mode shape to the post-buckling deflection was identified.

Panel flutter is a self-excited oscillation of a plate or a shell in supersonic flow on oneside. Because of aerodynamic loading on the panel, two Eigen modes of the structure mergeand lead to this dynamic instability (Xue and Mei, 1993). Supersonic flutter of plates andshells was recognized to be an important aspect of the design of high speed vehicles whenJordan (1956) observed that a number of the early V-2 rocket failures were due to panelflutter. Since then, extensive analytical and experimental research on that subject has beenperformed. A common remedy to the flutter problem is to stiffen those panels in danger offlutter, a method that usually introduces additional weight to the design.

Thin plates are a commonly used form of structural components especially in aerospacevehicles, such as high-speed aircraft, rockets, and spacecrafts, which are subjected to ther-mal loads due to aerodynamic and/or solar radiation heating. This results in a temperaturedistribution over the surface and a thermal gradient through the thickness of the plate. Thepresence of these thermal fields results in a flutter motion at a lower dynamic pressure, or alarger limit-cycle amplitude at the same dynamic pressure. In addition, a high temperaturerise may cause large thermal deflections (thermal buckling) of the skin panels, which couldaffect flutter response. Accordingly, it is important to consider the interactive effect of bothflutter and thermal buckling. However, it appears that in most past studies, the phenomenaof thermal buckling and nonlinear supersonic flutter were investigated separately especiallyin the field of adaptive and functionally graded materials.

Extensive research work has been carried out on the FGM since the concept was pro-posed in the late 1980s in Japan. FGMs are non-homogeneous composites characterized bya smooth and continuous change of material properties from one surface to the other. Thisis achieved by gradually varying the volume fraction of the constituent materials. One ofthe advantages of using these materials is that they can survive environments with high tem-perature gradients, while maintaining structural integrity. Functionally graded materials areusually composed of two or more materials whose volume fractions are changing smoothlyand continuously along desired direction(s). This continuous change in the compositionsleads to a smooth change in the mechanical properties, which has many advantages over thelaminated composites, where the delamination and cracks are most likely to be initiated at theinterfaces due to the abrupt variation in the mechanical properties between laminas. Dynamicanalyses on FGM plates were first done by a Japanese group led by Tani as early as the early1990s, and these works were reported in Tani and Liu (1993), Liu et al. (1991), Liu and Tani,1992, and Liu and Tani 1994. Dai et al. (2005) presented a mesh free model for the active

FUNCTIONALLY GRADED MATERIAL PLATES SUBJECT 2149

shape control and the dynamic response suppression of a functionally graded material platecontaining piezoelectric sensors and actuators. Birman (1997) studied the stability of func-tionally graded shape memory alloy (SMA) hybrid sandwich panels under the simultaneousaction of in-plane compressive and thermal loadings. Functional grading was achieved by anon-uniform distribution of shape memory alloy fibers in the middle plane (sinusoidal distri-bution). El-Abbassi and Meguid (2000) presented a new thick shallow shell element to studythe thermoelastic behavior of functionally graded structures made from shells and plates.The element accounts for the varying elastic and thermal properties across its thickness.Reddy (2000) presented theoretical formulation, Navier’s solutions of rectangular plates,and finite element model based on the third order shear deformation plate theory for analysisthrough the thickness functionally graded plates. The formulation accounts for the thermo-mechanical coupling, time dependence, and von Karman-type geometric nonlinearity. He etal. (2001) presented a finite element formulation based on the classical laminated plate the-ory for the shape and vibration control of functionally graded material plates with integratedpiezoelectric sensors and actuators. Javaheri and Eslami (2002) derived the equilibrium andstability equations of a rectangular plate made of a functionally graded material under ther-mal loads adopting the higher order shear deformation plate theory. A buckling analysis of afunctionally graded plate under four types of thermal loads is carried out and a closed formsolution for the prediction of the buckling temperature for rectangular simply supported FGMplate was obtained. Woo et al. (2003) developed an analytical solution for the post-bucklingbehavior of plates and shallow cylindrical shells made of functionally graded materials underthe simultaneous action of compressive in-plane loads and a temperature field. The solutionis obtained in terms of mixed Fourier series. Yang et al.(2004) investigated the geometricallynonlinear bending behavior of functionally graded plates with integrated piezoelectric layerssubjected to transverse loads and a temperature gradient through the plate thickness. Zenk-our (2005) studied the static response for a simply supported functionally graded rectangularplate subjected to a transverse uniform load using a generalized shear deformation theory.Kim (2005) developed an analytical technique to investigate the effect of temperature onthe vibration characteristics of thick functionally graded rectangular plates, taking into ac-count the temperature dependence of the material properties. Batra and Jin (2005) adoptedthe first-order shear deformation theory (FSDT) coupled with the finite element method tostudy the vibration of functionally graded anisotropic rectangular plate with different edgesupport conditions. The grading there was achieved through continuously changing the fiberorientation angle through the thickness. Prakash and Ganapathi (2006) studied the influenceof thermal environment on the critical flutter dynamic pressure of a flat functionally gradedpanel. The governing equations of motion are derived using Lagrange’s equations. Their for-mulation does not really capture the effect of the history of variation of the thermal expansioncoefficient with temperature. Navazi and Haddadpour (in press) analytically investigated theaero-thermoelastic stability margins of thin functionally graded panels. Haddadpour et al.(in press) studied the nonlinear aeroelastic behavior of thin functionally graded plates in su-personic flow. Ibrahim et al. (2006) investigated the thermal buckling and flutter boundariesof thin FGM plates at elevated temperature. The governing equations are derived using theprinciple of virtual work and the classical plate theory based on an incremental formulationto capture the effect of the history of variation of the thermal expansion coefficient withtemperature on the panel response.

2150 H. H. IBRAHIM and M. TAWFIK

In this work, the nonlinear flutter and the thermal post-buckling behavior of a ceramic-metal functionally graded plate under combined thermal and aerodynamic loads are studiedusing a nonlinear finite element method. The nonlinear governing equations of motion forthin rectangular functionally graded plate are obtained using the principle of virtual workand the von Karman strain-displacement relation. The approach is based on thermal strainbeing a cumulative physical quantity to take into account the temperature-dependence ofmaterial properties (Guo, 2005). The static nonlinear equations of motion are solved byNewton-Raphson numerical technique to get the thermal post-buckling deflection. The dy-namic nonlinear equations of motion are transformed to modal coordinates and the Newmarkimplicit integration scheme is employed to solve the second order ordinary differential equa-tions of motion. Numerical results are provided to show the effect of thermal field, dynamicpressure, and volume fraction exponent on the post-buckling and the limit-cycle oscillationof clamped functionally graded rectangular plates.

2. FINITE ELEMENT FORMULATION

The equation of motion with the consideration of large deflection and temperature depen-dence of material properties are derived for a functionally graded plate subject to aerody-namic and thermal loads. To account for temperature dependence of material properties,the thermal strain is modeled as an integral quantity of the thermal expansion coefficient.The element used in this study is the rectangular four-node-Bogner-Fox-Schmidt (BFC) C1

conforming element (for the bending DOFs) (Bogner et al, 1966).

2.1. Displacement-Nodal Displacement Relation

The nodal degrees of freedom vector {�} of the rectangular plate element can be written as:

��� �������

�x���

�y��2�

�x�y

�� �u� ��

�T

�� ��b���m�

�(1)

where {�b} is the transverse displacement vector and {�m} is the membrane displacementvector. The displacement-nodal displacement relation can be presented in terms of interpo-lation function matrices, [N� ], [Nu] and [N� ] as:

� � [N� ] ��b� e � u � [Nu] �u� e and � � [N� ] ��� e (2)

where the superscript, e, indicates element degrees of freedom.

2.2. Nonlinear Strain-displacement Relations

The inplane strains and curvatures, based on von Karman moderately large deflection andclassical thin plate theory, are given by:

FUNCTIONALLY GRADED MATERIAL PLATES SUBJECT 2151

��������x

�y

� xy

����� ������������������

�u

�x��

�y

�u

�y� ���x

����������������

�������������������

1

2

���

�x

2

1

2

���

�y

2

��

�x

��

�y

������������������ z

�������������������

��2�

�x2

��2�

�y2

�2�2�

�x�y

������������������ (3)

Or in compact form

��� � ��m� � ���� � z �� � (4)

Parameters u, � and � are displacements in x, y and z directions, respectively. �m , �� , and zare the membrane linear strain vector, the membrane nonlinear strain vector, and the bendingstrain vector, respectively.

2.3. Functionally Graded Materials

Typically, the FGMs are made of a mixture of two materials� a ceramic, which is capable ofwithstanding high temperature environments due to its low thermal conductivity and a metalthat acts as a structural element to support loading and prevent fractures. Without losinggenerality, it is usually assumed that the top surface of an FGM plate is ceramic rich andthe bottom is metal rich. The region between the two surfaces consists of a blend of thetwo materials which is assumed in the form of a simple power law distribution as (Dai et al,2005):

Pe z� � PC VC � PM 1� VC� (5)

VC ��

0�5� z

h

�n� �h�2 � z � h�2� 0 � n � �� (6)

where z is coordinate in the thickness direction of a plate� (Pe� Pc� PM) are effective materialproperties of the FGM, the properties of the ceramic and the properties of the metal respec-tively, Vc is the ceramic volume fraction and h is the plate thickness. Power n is the volumefraction exponent.

Figure 1 shows the variation of the volume fraction function versus the non-dimensionalthickness, z/h, with different volume fraction exponent n.

2.4. Stress-strain Relationship of an FGM Panel

The relationship of inplane forces {N} and bending moments {M} in terms of the strainvector can be written as (Guo, 2005):�

N

M

���

A B

B D

���

���

N T

M T

�(7)

2152 H. H. IBRAHIM and M. TAWFIK

Figure 1. Variation of the ceramic volume fraction function versus the non-dimensional thickness z/h.

where

A� B� D� �h�2�

�h�2

1� z� z2� [Qz� T �] dz

Qz� T � �

����������

Ez� T �

1� �2z�

�Ez� T �

1� �2z�0

�Ez� T �

1� �2z�

Ez� T �

1� �2z�0

0 0Ez� T �

2 1� �z��

����������

�N T �� �M T �� �h�2�

�h�2

���� T�Tref

[Qz� ��] ����� d�

���� 1� z� dz

where [A], [B] and [D] are the extensional matrix, extensional-bending coupling matrix andflexural matrix, respectively. T denotes the temperature, while a constant temperature distri-bution in the x, y and z directions are assumed.

Substituting equation (4) into (7) yields

FUNCTIONALLY GRADED MATERIAL PLATES SUBJECT 2153

�N � � [A] ��m � ��� � [B]�� � �N T � (8)

�M� � [B] ��m � ��� � [D]�� � �M T � � (9)

2.5. Aerodynamic Loading

The first-order quasi-steady piston theory for supersonic flow states that (Tawfik et al, 2002):

Pa � ��

ga

�o

D11

a4

��

�t� �D11

a3

��

�x

(10)

with

q � �a�2

2� � �

�M2� � 1� �o �

�D11

�ha4

12

� Ca ��

M2� � 2

�2�M2� � 1

�2

�aa

�h�

ga � �a��

M2� � 2

��h�o�

3 � �Ca and � � 2qa3

�D11

where Pa is the aerodynamic loading, � is the velocity of airflow, M� is the Mach number,q is the dynamic pressure, �a is the air mass density, Ca is the aerodynamic damping co-efficient, ga is non-dimensional aerodynamic damping, � is non-dimensional aerodynamicpressure, D11 is the first entry of the flexural stiffness matrix D1� 1� and a is the panellength.

2.6. Governing Equations

By using the principle of virtual work and equations (2), (4), (8) and (9), the governingequation of thermal post-buckling and nonlinear panel flutter of a functionally graded platecan be derived as follows

�W � �Wint � �Wext � 0� (11)

The internal virtual work �Wint is given as

�Wint ��A

����m�T �N � � ���t �M��

d A

� ����T�

[k]� [kT ]� 1

2[n1]� 1

3[n2]

��� � ����T �pT � (12)

2154 H. H. IBRAHIM and M. TAWFIK

where ��� � [� �x � y �xy u �] is the displacement vector� [k] and [kT ] are the linearstiffness matrix and the thermal geometric stiffness matrix� [n1] and [n2] are the first- andsecond-order nonlinear stiffness matrices, respectively. In addition, {pT } is the thermal loadvector.

On the other hand, the virtual work done by external forces on a plate element, consid-ering inertia and random pressure excitations, is:

�Wext ��A

�� ��h � � Pa�� �u ��h u�� �� ��h ��� d A� (13)

The virtual work of aerodynamic loading, Pa , using the quasi-steady first-order piston theorycan be written as:�

A

�� Pad A � ��A

��b

�ga

�o

D11

a4

��b

�t� �D11

a3

��b

�x

d A

� ����b�T!g" � �b� � ���b�T � [aa] ��b� � (14)

The finite element form of the virtual work done by inertia and aerodynamic forces on a plateelement takes the form:

�Wext � ����b

��m

�T �mb 0

0 mm

�� �b

�m

�����b

��m

�T �g 0

0 0

�� �b

�m

�

� �

���b

��m

�T �aa 0

0 0

���b

�m

�(15)

where [mb] and [mm] are bending and inplane inertia matrices, respectively. [g] is the aero-dynamic damping matrix, and [aa] is the aerodynamic influence matrix.

Combining equations (12) and (15), the element equation of motion is expressed as:

[m]# �$� !g" # �$��� [aa]� [k]� [kT ]� 1

2[n1]� 1

3[n2]

��� � �pT � � (16)

Assembling the element equations of motion to the system level by summing up the con-tributions from all elements and applying the boundary conditions, the system equations ofmotion become:

[M]# W$� [G]

# W$��� [Aa]� [K ]� [KT ]� 1

2[N1]� 1

3[N2]

�W � � �PT � � (17)

Equation (17) is a set of nonlinear differential equations describing the panel motion undercombined thermal and aerodynamic loads.

FUNCTIONALLY GRADED MATERIAL PLATES SUBJECT 2155

3. SOLUTION PROCEDURES

3.1. Thermal Post-buckling Deflection

The physical problems associated with the system equations of motion can be categorizedinto two types: static and dynamic problems. For the static thermal problem, inertia term isdropped as well as the terms related to the aerodynamic load. Therefore, for static thermalbuckling problem, equation (17) becomes:�

[K ]� [KT ]� 1

2[N1]� 1

3[N2]

�W � � �PT � � (18)

Introducing the function ��W �� to equation (18),

�� W �� ��

[K ]� [KT ]� 1

2[N1]� 1

3[N2]

�W � � �PT � � 0� (19)

This can be written using truncated Taylor expansion as follows:

�� W � �W �� � �� W �� � d �� W ��dW �

��W � �� 0 (20)

where

d �� W ��dW �

� [K ]� [KT ]� [N1]� [N2]� � [Ktan] � (21)

Thus, the Newton-Raphson iterative procedures for the determination of the post-bucklingdeflection can be expressed as follows:

�� W ��i ��

[K ]� [KT ]� 1

2[N1]i � 1

3[N2]i

�W � � �PT � (22)

[Ktan]i ��W �i�1 � ��� W ��i (23)

��W �i�1 � � [Ktan]�1i �� W ��i (24)

�W �i�1 � �W �i � ��W �i�1 � (25)

Convergence occurs in the above procedure when the maximum value of ��W �i�1 becomesless than a given tolerance �tol � i.e. max ���W �i�1� � �tol.

3.2. Dynamic Response using Modal Coordinates

The plate could demonstrate four types of flutter limit-cycle oscillation: nearly harmonicoscillation, periodic oscillation, non-periodic oscillation, and chaotic oscillation. For theflutter limit-cycle oscillation with thermal effect, equation (17) can be stated as:

2156 H. H. IBRAHIM and M. TAWFIK

�Mb 0

0 Mm

�� Wb

Wm

���

G 0

0 0

�� Wb

Wm

�

�%�

�Aa 0

0 0

���

Kb Kbm

Kmb Km

���

KT b 0

0 0

�

� 1

2

�N1Nm �Wm��� N1Nb �Wb�� N1bm

N1mb 0

�� 1

3

�N2b 0

0 0

�&�Wb

Wm

�

��

PT b

PT m

�(26)

where the subscripts m and b denote membrane and bending, respectively. Separating themembrane and transverse displacement equations in equation (26):

[Mb]# Wb

$� [G]# Wb

$��� [Aa]� [Kb]� [KT b]� 1

2[N1Nm �Wm��]

� 1

2[N1Nb �Wb��]� 1

3[N2b]

�Wb� �

�[Kbm]� 1

2[N1bm]

�Wm� � �PT b� (27)

[Mm]# Wm

$��[Kmb]� 1

2[N1mb]

�Wb� � [Km] �Wm� � �PT m� � (28)

Note that, neglecting the in-plane inertia term will not bring significant error, because in-plane natural frequencies are usually a 2–3 order of magnitude higher than bending ones.Therefore, the in-plane displacement can be expressed in terms of bending displacement as:

�Wm� � [Km]�1

��PT m� �

�[Kmb]� 1

2[N1mb]

�Wb�

� [Km]�1 �PT m� � [Km]�1 [Kmb] �Wb� � 1

2[Km]�1 [N1mb] �Wb�

� �Wm�o � �Wm�1 � �Wm�2 � (29)

It can be shown that (Xue, 1991):

1

2[N1bm] �Wm�o � 1

2

!N1Nm �Wm�o�

" �Wb� � (30)

Accordingly, by substituting equation (29) into (27), equation (27) can be written as:

FUNCTIONALLY GRADED MATERIAL PLATES SUBJECT 2157

[Mb]# Wb

$� [G]# Wb

$��� [Aa]� [Kb]� [KT b]� [Kbm] [Km]�1 [Kmb]

� !N1Nm �Wm�o�

"� 1

2[N1Nb]� 1

2

!N1Nm �Wm�1�

"� 1

2[N1bm] [Km]�1 [Kmb]

� 1

2[Kbm] [Km]�1 [N1mb]� 1

3[N2b]� 1

4[N1bm] [Km]�1 [N1mb]

� 1

2

!N1Nm �Wm�2�

" �Wb�

� �PT b� � [Kbm] [Km]�1 �PT m� � �Pb� � (31)

An effective solution procedure is to transform equation (31) into modal coordinates usingreduced system normal modes by expressing the system bending displacement �Wb} as alinear combination of some mode shapes as:

�Wb� n'

r�1

qr ��r � � [�] �q� (32)

where the r th normal mode ��r � and the corresponding natural frequency �r are obtainedfrom the linear vibration of the system as:

�2r [Mb] ��r � �

�[Kb]� [Kbm] [Km]�1 [Kmb]

� ��r� � (33)

Based on the normal modes evaluated in equation (33), all the matrices in equation (31) aretransformed into modal coordinates. Accordingly, equation (31) can be written in modalcoordinates as:! �Mb

" �q� � �2 !� r�r

" ! �Mb

"� ! �G"� �q� � �! �K "� ! �Kq

"� ! �Kqq

"� �q� � # �Pb

$(34)

where the modal mass and modal linear stiffness matrices are given by:�! �Mb

"�! �G" � ! �K "� � [�]T [Mb] � [G] � [Klin]� [�] (35)

and

[Klin] � � [Aa]� [Kb]� [KT b]� [Kbm] [Km]�1 [Kmb]� !N1Nm �Wm�o�"� (36)

The first-order and second-order nonlinear modal stiffness matrices are given by:

! �Kq

" � [�]Tn'

r�1

qr

�1

2[N1Nb]r� � 1

2

!N1Nm �Wm�1�

"r�� 1

2[N1bm]r� [Km]�1 [Kmb]� 1

2[Kbm] [Km]�1 [N1mb]r�

[�] (37)

2158 H. H. IBRAHIM and M. TAWFIK

Table 1. Temperature-dependence coefficients for nickel and silicon nitride.

Properties Material P�1 Po P1 P2 P3

E (MPa) Si3N4 0 348.43e9 –3.07e-4 2.2e-7 –8.9e-11Nickel 0 223.95e9 –2.79e-4 3.9e-9 0

� Si3N4 0 0.24 0 0 0Nickel 0 0.31 0 0 0

� Si3N4 0 5.8723e-6 9.09e-4 0 0Nickel 0 9.9209e-6 8.71e-4 0 0

� Kg/m3� Si3N4 2370Nickel 8900

! �Kqq

" � [�]Tn'

r�1

n's�1

qr qs

�1

3[N2b]rs� � 1

2

!N1Nm �Wm�2�

"rs�

� 1

4[N1bm]r� [Km]�1 [N1mb]s�

[�] (38)

# �Pb

$ � [�]T �Pb� � (39)

A modal structural damping matrix 2 [� r �r ] [Mb] has been added to equation (34) to accountfor the structural damping effect on the system. The coefficient � r is the modal damping ratioof the r th mode, while �r is the r th natural frequency in Hz.

4. NUMERICAL RESULTS AND DISCUSSIONS

4.1. Thermal Buckling and Post-buckling of an FGM Panel

In this section, thermal buckling and post-buckling analysis are carried out for an FGM panel,which is a mixture of nickel and silicon nitride (Si3N4) to figure out the effect the volumefraction exponent, n, and the different boundary conditions on the buckling characteristics ofthe FGM panel. The material properties are assumed to be temperature dependent accordingto the following relation (Touloukian, 1976):

P � Po

�P�1T�1 � 1� P1T � P2T 2 � P3T 3

�� (40)

The coefficients Po, P�1, P1, P2 and P3 for Young’s modulus E, the Poisson ratio � and thethermal expansion coefficient � of nickel and silicon nitride are given in Table 1.

To validate the present formulation, results were compared to an analytical solution(Paul, 1982) that was derived using 25-term series solution for an aluminum square platewith all edges clamped and dimensions 0.305 � 0.305 � 0.002 m. Table 2 presents the re-sulting maximum deflection to thickness ratio obtained by different models at a temperaturerise value of 9.67 �C.

FUNCTIONALLY GRADED MATERIAL PLATES SUBJECT 2159

Figure 2. Post-buckling deflection for a clamped FGM panel with different volume fraction exponent n.

Table 2. Resulting Wmax/h compared to classical solution at a temperature rise value of9.67 �C.

Classical 4 Elements 16 Elements 36 Elements 64 Elements 144 Elements(Paul, 1982)1 0.84 0.91 0.96 0.985 0.99

The geometry of the panel adopted hereinafter is chosen to be 0.305� 0.305� 0.002 m.A 8 � 8 mesh is utilized, as solution convergence is found to occur at this mesh size.

Figure 2 illustrates the effect of the volume fraction exponent on the buckling charac-teristics of a clamped FGM panel. It is seen that the responses that correspond to propertiesintermediate to that of the metal and the ceramic lie between that of the metal and ceramic,which is consistent with Reddy’s observations (Reddy, 2000).

Figure 3 shows the thermal buckling of functionally graded material panels with simplysupported edges, which are immovable in both x- and y- directions. It is seen that there isno buckling phenomenon because any small temperature rise results in a prompt transversedeflection of the panel due to structural asymmetry about the middle plane of the FGM panelthat makes all simply supported FGM panels lose their buckling phenomena.

Although the ceramic panel response seems to be superior to that of the FGM panel,using the ceramic panel results in a structural integrity problem due to the brittleness of theceramic. Therefore, increasing the ceramic volume fraction is bounded by structural integrityaspects.

2160 H. H. IBRAHIM and M. TAWFIK

Figure 3. Post-buckling deflection for a simply supported FGM panel with different volume fractionexponent n.

4.2. Flutter Limit-cycle Oscillations

Limit-cycle oscillation (LCO) can be categorized into four types: nearly harmonic LCO,periodic LCO, non-periodic oscillation, and chaotic oscillation. This section presents timedomain solution using modal transformation to predict the vibration time history of all typesof panel flutter. Plates made from nickel, silicon nitride (Si3N4), and nickel-silicon nitridefunctionally graded material are studied and compared. Time history responses at the max-imum deflection points of the panel are obtained for a nickel-silicon nitride functionallygraded material plate with a unity volume fraction exponent (n � 1) and at various combina-tions of temperature rises ( T) and non-dimensional dynamic pressure (�). A proportionaldamping ratio of � r �r � � s �s is used with fundamental modal damping coefficient � 1 equalto 0.02. The aerodynamic damping coefficient Ca is set to 0.1. Newmark implicit numericalintegration scheme (Bathe, 1996) is utilized to solve the system differential equations witha time step of integration equal to 1/10,000 sec., while Newton-Raphson iteration scheme isadopted to solve the nonlinear algebraic system of equations at each time step. Seven modeswith an 8 � 8 mesh are found to be adequate. The seven mode shapes utilized in modaltransformation are shown in Figure 4.

A nickel-silicon nitride FGM panel under the effect of thermal and aerodynamic load-ing is presented in Figure 5 in terms of the critical temperature boundary and linear flutterboundary (Ibrahim et al., 2006). D11 is evaluated at Tref and n � 8 (i.e., D11 is evaluatedusing the nickel properties at room temperature). The area of the graph is divided into threeregions� the flat panel region, in which the panel is stable, i.e. neither buckling nor panel

FUNCTIONALLY GRADED MATERIAL PLATES SUBJECT 2161

Figure 4. The seven mode shapes adopted in modal transformation.

Figure 5. The effectiveness of the volume fraction exponent n on the stability boundaries of an FGMclamped panel.

flutter occurred� the buckled region, in which the thermal stresses overcome the panel stiff-ness and aerodynamic stiffness. In this region, the panel undergoes static instability underinplane thermal loading� the third region is the flutter region, where the panel undergoesdynamic instability under the influence of aerodynamic pressure. Thus, the wider the flatpanel region is, the more stable the panel is. It is seen in Figure 5 that decreasing the volumefraction exponent n, results in a wider flat panel region and in turn a more stable panel.

Figure 6 presents a comparison between the limit cycle oscillations of nickel, siliconnitride, and nickel-silicon nitride FGM plates at a temperature rise of 10 �C and a non-

2162 H. H. IBRAHIM and M. TAWFIK

Figure 6. Comparison between the limit-cycle oscillations of nickel, Si3N4 and nickel-Si3N4 FGM platesat T = 10 �C and � = 850.

dimensional dynamic pressure � � 850. It is seen that, at high dynamic pressures and lowtemperature rises, the displacement time histories exhibit nearly harmonic flutter oscillation.It is also seen that the ceramic plate has lower limit-cycle amplitude than the nickel plate andthe flutter amplitude that corresponds to properties intermediate to that of the metal and theceramic lie between that of the metal and ceramic.

The four types of limit-cycle flutter oscillations of a functionally graded nickel-siliconnitride plate with volume fraction exponent n equals one are shown in Figures 7–10, repre-sented by time history and phase plots. At � � 850 and T = 10 �C, as shown in Figure 7,the phase plot and time history clearly indicate a nearly harmonic limit-cycle flutter oscil-lation. With increasing T value until the thermal buckling boundary crossing shown inFigure 5 while decreasing the dynamic pressure �, the nearly harmonic LCO has the ten-dency to evolve into periodic LCO as the deflection-dependent nonlinear stiffness terms,which is added to the system by increasing the temperature beyond the critical value, startsto interact with the flutter oscillation resulting in a distortion in the periodicity of the flutteroscillations. Therefore, it is found that the periodic LCO region resides in the lower right ofthe nearly harmonic LCO region, as shown in Figure 8 where � � 500 and T = 25 �C. Byincreasing T further, the plate motion evolves from periodic to non-periodic oscillation, asshown in Figure 9, where � � 500, and T = 40 �C. At a combination of a medium dynamicpressure and a high temperature rise, such as � � 400, and T = 50 �C, the plate is likelyto undergo chaotic oscillation, as shown in Figure 10. Compared with non-periodic oscilla-tion, chaos shows diffusion on the phase plot, which fills most of the region from minima tomaxima.

FUNCTIONALLY GRADED MATERIAL PLATES SUBJECT 2163

Figure 7. Nearly harmonic LCO of a clamped nickel-Si3N4 FGM plate with n = 1, T = 10 �C and� = 850.

Figure 8. Periodic LCO of a clamped nickel-Si3N4 FGM plate with n = 1, T = 25 �C and � = 500.

2164 H. H. IBRAHIM and M. TAWFIK

Figure 9. Non-periodic LCO of a clamped nickel-Si3N4 FGM plate with n = 1, T = 40 �C and � = 500.

Figure 10. Chaotic oscillation of a clamped nickel-Si3N4 FGM plate with n = 1, T = 50 �C and � = 400.

FUNCTIONALLY GRADED MATERIAL PLATES SUBJECT 2165

5. CONCLUSIONS

A nonlinear finite element formulation is presented for the analysis of thermal buckling andnonlinear aero-thermal flutter response of an FGM panel made of nickel and silicon nitride.Nonlinear temperature-dependent material properties and von Karman large deflection wasconsidered in the formulation. The material properties were assumed to vary through thethickness direction based on a simple power law distribution. The aerodynamic pressureis modeled using the quasi-steady first-order piston theory. An approach based on thermalstrain being a cumulative physical quantity was adopted to take into account the temperaturedependence of material properties.

The effectiveness of the volume fraction exponent and the boundary conditions on thethermal buckling characteristics of the FGM panel is studied. The results showed thatthe presence of the silicon nitride with the nickel enhances the buckling characteristics ofthe panel through increasing the buckling temperature and decreasing the post-bucklingdeflection.

A time domain solution is presented to numerically investigate the flutter limit-cycleoscillations at elevated temperatures of a clamped FGM panel. The finite element modalformulation and solution procedures are developed for the time domain method. Resultsshowed that the ceramic plate has lower flutter limit-cycle amplitude than the nickel plateand the flutter amplitude that corresponds to properties intermediate to that of the metal andthe ceramic lie between that of the metal and the ceramic. It is also concluded that a desiredflutter limit-cycle oscillation type and amplitude can be reached through tuning the value ofthe volume fraction exponent taking into account the structural integrity aspects.

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