Proceedings of ICTACEM 2007 International Conference on Theoretical, Applied, Computational and Experimental Mechanics December 27-29, 2007, IIT Kharagpur, India

ICTACEM-2007/336

Dynamic Instability of Functionally Graded Cylindrical Panels S. Pradyumna 1 and J.N. Bandyopadhyay 2 Department of Civil Engineering, Indian Institute of Technology, Kharagpur, West Bengal, India 721302

ABSTRACT In the present paper, dynamic instability behavior of functionally graded (FG) cylindrical panels subjected to inplane pulsating load is studied using the modified first-order shear deformation theory in conjunction with the finite element approach. The panels are subjected to thermal field and the material properties are assumed to be temperature dependent and graded in the thickness direction according to the power-law distribution in terms of volume fraction of the constituents. The boundaries of instability regions are obtained using the principle of Bolotin’s method. Keywords: Dynamic instability; functionally graded material; finite element method; cylindrical panels.

1. INTRODUCTION Functionally graded materials (FGMs) are microscopically heterogeneous composites usually made from a mixture of metals and ceramics. The material properties of FGM are spatially varied and are controlled by the variation of the volume fraction of the constituent materials. FGMs have the advantages of heat and corrosion resistance typical of ceramics and mechanical strength and toughness typical of metals. In view of these, there is an increased interest among researchers to study the dynamic behavior of the structural components made of these materials. Functionally graded materials are now being strongly considered as a potential structural material for future high-speed spacecraft. Though intensive researches were reported in the available literature on the FGMs, investigations on the dynamic and stability behaviors of FGM shell panels are still limited. Among the available literature, Birman1 studied the buckling of FG hybrid composite plates. Loy et al.2 studied the free vibration of simply supported FG cylindrical shells. Pradhan et al.3 investigated the effect of different boundary conditions on the natural frequencies of FG cylindrical shells made up of stainless steel and zirconia. Influence of the volume fractions and the effects of configurations of the constituent materials on the parametric instability regions of FG plates were investigated by Ng et al.4. Dynamic stability of simply supported 1 2

Research scholar (corresponding author), Phone: +91-3222-281415; Email- [email protected] Professor, Phone: +91-3222-283404; Email- [email protected]

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FG cylindrical shells under harmonic axial loading was analyzed by Ng et al.5 using a normal mode expansion and Bolotin’s method to determine the boundaries of instability. Yang and Shen6 investigated the free and forced vibration characteristics of shear deformable initially stressed FG plates in thermal environment. By considering the material properties of the constituents to be nonlinear functions of the temperature and graded in the thickness direction by power law distribution, Yang and Shen7 investigated the free vibration and parametric resonance of shear deformable FG cylindrical panels. Patel et al.8 carried out free vibration analysis of FG elliptical and cylindrical shells using a higher-order theory. Nonlinear free vibration behavior of FG plates is investigated by Woo et al.9. By using moving least squares differential quadrature method, Lanhe et al.10 investigated the dynamic stability of thick FG plates subjected to aero-thermo-mechanical loads. To the best of authors’ knowledge, no published results are available on the dynamic instability regions of FGM cylindrical panels subjected to uniformly distributed in-plane load and temperature field. In the present paper, dynamic instability behavior of FG shell panels subjected to in-plane pulsating loads and thermal field is investigated. Finite element formulation based on the modified first-order shear deformation theory as proposed by Tanov and Tabiei11 is used to carry out the analyses. The formulation presented here results in parabolic distribution for both strains and stresses and, therefore, the use of shear correction factor is eliminated. The dynamic instability problem is solved using the Bolotin’s approach. The accuracy of the formulation is validated by comparing the results with those available in the existing literature. Further, investigation is carried out to study the effects of boundary conditions, volume fraction index and radius of curvature on the dynamic instability of the FGM cylindrical panels. 2. THEORETICAL FORMULATIONS Let us consider a shell element made of a FG material with the co-ordinate system (x, y, z) shown in Fig.1, and chosen such that the plane x-y at z = 0 coincides with the mid-plane. In order to approximate the three-dimensional elasticity problem to a two-dimensional one, the displacement components u(x,y,z), v(x,y,z) and w(x,y,z) at any point in the shell space are expanded in Taylor’s series in terms of the thickness co-ordinates. The elasticity solution indicates that the transverse shear stresses vary parabolically through the thickness. This requires the use of a displacement field in which the in-plane displacements are expanded as cubic functions of the thickness co-ordinate. The displacement fields, which satisfy the above criteria, are assumed in the form as,

2

u = u0 + zθ y + z 2ϕ y + z 3ψ y v = v0 − zθ x − z 2ϕ x − z 3ψ x

(1)

w = w0 where u, v and w are the displacements of a general point (x, y, z) in an element of the laminate along x, y and z directions, respectively. The parameters u0, v0, w0, θ x and θ y are the displacements and rotations of the middle plane, while ϕx , ϕ y , ψ x and ψ y are the higherorder displacement parameters defined at the mid-plane. Using the strain-displacement relations 1 ⎛ ∂ui ∂u j + 2 ⎝ ∂x j ∂xi

ε ij = ⎜ ⎜

⎞ ⎟⎟ ⎠

(2)

the components of the strain vector corresponding to the displacement field (1) are

εx =

∂ϕ y 2 ∂ψ y 3 ∂u0 ∂θ y + z+ z + z ∂x ∂x ∂x ∂x

εy =

∂v0 ∂θ x ∂ϕ ∂ψ x 3 − z − x z2 − z ∂y ∂y ∂y ∂y

⎛ ∂u ∂v ⎞ ⎛ ∂θ y ∂θ x ⎞ ⎛ ∂ϕ y ∂ϕ x ⎞ 2 ⎛ ∂ψ y ∂ψ x − − − 2ε xy = ⎜ 0 + 0 ⎟ + ⎜ ⎟z+⎜ ⎟z +⎜ ∂x ⎠ ⎝ ∂y ∂x ⎠ ∂x ⎝ ∂y ∂x ⎠ ⎝ ∂y ⎝ ∂y

⎞ 3 ⎟z ⎠

(3)

⎛ ∂w ⎞ 2ε yz = ⎜ 0 − θ x ⎟ − 2ϕ x z − 3ψ x z 2 ⎝ ∂y ⎠

⎛ ∂w ⎞ 2ε xz = ⎜ 0 + θ y ⎟ + 2ϕ y z + 3ψ y z 2 x ∂ ⎝ ⎠ Vanishing of the transverse shear stresses at the top and bottom shell surfaces,

σ yz ( ± h /2) = σ xz ( ± h /2) = 0 , makes the corresponding strains zero, which yields, ϕx = ϕ y = 0 , ψ x =

4 3h 2

⎞ 4 ⎛ ∂w ⎛ ∂w0 ⎞ − θ x ⎟ and ψ y = 2 ⎜ 0 + θ y ⎟ ⎜ 3h ⎝ ∂y ⎝ ∂x ⎠ ⎠

in which h is the shell thickness. Using these expressions, Eqs. (1) and (3) are simplified as

u = u0 + zθ y + z 3ψ y v = v0 − zθ x − z 3ψ x

(4)

w = w0 and

εx =

∂ψ y 3 ∂u0 ∂θ y + z+ z ∂x ∂x ∂x

3

εy =

∂v0 ∂θ x ∂ψ x 3 − z− z ∂y ∂y ∂y

⎛ ∂u ∂v 2ε xy = γ xy = ⎜ 0 + 0 ⎝ ∂y ∂x

⎞ ⎛ ∂θ y ∂θ x − ⎟+⎜ ∂x ⎠ ⎝ ∂y

⎞ ⎛ ∂ψ y ∂ψ x − ⎟z+⎜ ∂x ⎠ ⎝ ∂y

⎞ 3 ⎟z ⎠

(5)

⎛ ∂w ⎞⎛ 4 ⎞ 2ε yz = γ yz = ⎜ 0 − θ x ⎟ ⎜ 1 − 2 z 2 ⎟ h ⎠ ⎝ ∂y ⎠⎝

4 ⎞ ⎛ ∂w ⎞⎛ 2ε xz = γ xz = ⎜ 0 + θ y ⎟ ⎜ 1 − 2 z 2 ⎟ h ⎠ ⎝ ∂x ⎠⎝ By assuming that the first two terms in the ε x , ε y and ε xy expressions in Eq. (5) represent the distribution of the in-plane strains through the thickness with enough accuracy. This means that we can neglect the contribution of the derivatives of ψ x and ψ y with respect to x and y and simplify the strain expressions as,

εx =

∂u0 ∂θ y + z ∂x ∂x

εy =

∂v0 ∂θ x − z ∂y ∂y

⎛ ∂u ∂v ⎞ ⎛ ∂θ y ∂θ x ⎞ − γ xy = ⎜ 0 + 0 ⎟ + ⎜ ⎟z ∂x ⎠ ⎝ ∂y ∂x ⎠ ⎝ ∂y

(6)

⎛ ∂w ⎞ 4 γ = ⎜ 0 − θ x ⎟ ⎜⎛ 1 − 2 z 2 ⎟⎞ h ⎠ ⎝ ∂y ⎠⎝ yz

∂w0 4 ⎞ ⎞⎛ + θ y ⎟ ⎜ 1 − 2 z2 ⎟ h ⎠ ⎝ ∂x ⎠⎝

γ = ⎛⎜ xz

These expressions are identical to the strain expressions from the first-order shear deformation displacement field except for the transverse shear strain expressions. 3. FG MATERIAL PROPERTIES The panels considered in the present analysis are assumed to be of uniform thickness h. It is also assumed that the panels are made from a mixture of ceramics and metals and the material composition is continuously varied such that the top surface (z = h/2) of the panels is ceramic rich, whereas the bottom surface (z = -h/2) is metal rich. Figure 2 shows the variation of volume fraction of FG material through the non-dimensional thickness of the panels. Thus, the effective material property P (such as Young’s modulus E, Poisson’s ratio ν , mass

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density ρ , coefficient of thermal expansion η , thermal conductivity K etc.) can be expressed as k

P = ∑ PjV j

(7)

j =1

where Pj and V j are the material property and volume fraction of the constituent material j, satisfying the volume fraction of all the constituent materials as, k

∑V j =1

j

=1

(8)

For a panel with the reference surface at its middle surface, the volume fraction can be written as

⎛ 2z + h ⎞ Vj = ⎜ ⎟ ⎝ 2h ⎠

n

(9)

in which, n characterizes the material variation through the panel thickness and 0 ≤ n ≤ ∞ . Also, all the material properties of the constituent materials such as, E , η and ν , which are temperature dependent, are assumed as, 2 3 P = P0 ( P−1 / T + 1 + PT ) 1 + P2T + PT 3

(10)

Non dim ensional thickness (z/h )

0.5

h

z

Ceramic

y

b

x Metallic

a

0.3 n=0 n=0.02

0.1

n=0.2 n=0.5

-0.1

n=1.0 n=2.0

-0.3

n=5 n=∞

-0.5

R Figure 1. Geometry of cylindrical FG panel

0

0.2

0.4

0.6

0.8

1

Volume fraction V

Figure 2. Variation of volume fraction along the thickness

where Pi (i = −1, 0,1, 2,3) are the coefficients of temperature T (K) and are unique for each constituent. Material properties for a FG solid with two constituent materials are given by, ⎛ 2z + h ⎞ P ( z ) = (Pc − Pm ) ⎜ ⎟ + Pm ⎝ 2h ⎠ n

(11)

5

where Pc and Pm refer to the corresponding properties of the ceramic and metal constituents, respectively. By using these material properties, the stresses can be determined as, ⎧σ x ⎫ ⎡ Q11 ⎪σ ⎪ ⎢Q y 12 ⎪⎪ ⎪⎪ ⎢ ⎢ τ = 0 ⎨ xy ⎬ ⎪τ ⎪ ⎢ 0 ⎪ yz ⎪ ⎢ ⎩⎪τ xz ⎭⎪ ⎢⎣ 0

Q12

0

0

Q22

0

0

0

Q66

0

0

0

Q44

0

0

0

0 ⎤ ⎛ ⎧ ε x ⎫ ⎧1 ⎫ ⎞ ⎜ ⎟ 0 ⎥⎥ ⎜ ⎪⎪ ε y ⎪⎪ ⎪1 ⎪ ⎟ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎥ ⎜ ⎨γ xy ⎬ − ⎨0 ⎬ ⎟ η ( z ) T ( z ) ⎟ ⎥⎜ 0 ⎥ ⎜ ⎪γ yz ⎪ ⎪0 ⎪ ⎟ ⎪ ⎪ ⎪ ⎪ Q55 ⎦⎥ ⎜⎝ ⎩⎪γ xz ⎭⎪ ⎪⎩0 ⎪⎭ ⎟⎠

(12)

where η ( z ) , T ( z ) are coefficient of linear thermal expansion and temperature at the stress free state, respectively, and

Q11 = Q22 =

E( z) ν E( z) E( z) ; Q12 = ; Q44 = Q55 = Q66 = G ( z ) = 2 2 2(1 +ν ) 1 −ν 1 −ν

(13)

Equation (12) can also be written as, {σ } = [Q]{ε }

(14)

where [Q ] is effective stiffness coefficient matrix given by the relation, ⎛ 2z + h ⎞ [Q] = (Qc − Qm ) ⎜ ⎟ + Qm ⎝ 2h ⎠ n

(15)

Qc and Qm are the effective stiffness coefficient matrices for ceramic and metal constituents, respectively. Also, for the value of the power n = 0, the FG panel becomes pure ceramic and metal for the value n = ∞ . Integrating the stresses through the thickness, the resultant forces and moments acting on the panel are obtained. ⎡ Nx ⎢ ⎢ Ny ⎢ N xy ⎣ or

⎡ N xT M x ⎤ h / 2 ⎡σ x ⎤ ⎢ ⎥ ⎢ ⎥ M y ⎥ = ∫ ⎢σ y ⎥ [1, z ] dz − ⎢ N Ty ⎢ 0 M xy ⎥⎦ − h / 2 ⎢⎣τ xy ⎥⎦ ⎣

M xT ⎤ ⎡ Qx ⎤ h / 2 ⎡τ xz ⎤ ⎥ M Ty ⎥ , [Q] = ⎢ ⎥ = ∫ ⎢ ⎥dz ⎣Qy ⎦ − h / 2 ⎣τ yz ⎦ 0 ⎥⎦

(16)

σ = Dε , where,

σ = ( Nx , Ny , Nxy , Mx , My , Mxy ,Qx ,Qy )

T

ε = (εx0 ,ε y0 ,γ xy0 ,κx ,κy ,κxy ,ϕx ,ϕy )

(17)

⎡ Aij ⎢ and [ D ] = ⎢ Bij ⎢0 ⎣

(18)

T

Bij Dij 0

0⎤ ⎥ 0⎥ Sij ⎥⎦

6

where Aij, Bij, Dij and Sij are the extensional, bending-stretching coupling, bending and transverse shear stiffnesses. They are defined as, h/2

Aij =

∫

h/2

Q ij dz , Bij =

−h / 2

∫

h/2

Q ij .z.dz , Dij =

−h / 2

∫

Q ij .z 2 dz.; i, j = 1, 2, 6

−h / 2

h/2

and Sij =

∫

−h / 2

Q ij .dz; i, j = 4, 5

Thermal force and moment resultants are given by, h/2

N xT = N Ty =

∫

{β } T ( z ) dz and M xT = M Ty =

−h / 2

h/2

∫

{β }T ( z ) z dz

(19)

−h / 2

⎧ (Q11 + Q12 )α ( z ) ⎫ ⎪⎪ ⎪⎪ in which, {β } = ⎨(Q12 + Q 22 )α ( z ) ⎬ ⎪ ⎪ 0 ⎩⎪ ⎭⎪

(20)

For a functionally graded panel, the temperature change through the thickness is not uniform, and is governed by the one-dimensional Fourier equation of heat conduction,

d ⎡ dT ⎤ =0 K ( z) ⎢ dz ⎣ dz ⎥⎦

(21)

subjected to conditions, T = Tc (at z = h / 2 ), T = Tm (at z = − h / 2 ) The solutions of Eq. (21) are obtained by means of polynomial series (Lanhe et al.10),

T ( z ) = Tm +

Tcm

ξ

5

where, ξ = ∑ j =0

5

∑ j =0

j

1 ⎛ K cm ⎞ ⎛ 2 z + h ⎞ ⎜− ⎟ ⎜ ⎟ jn + 1 ⎝ K m ⎠ ⎝ 2h ⎠

jn +1

j

1 ⎛ K cm ⎞ ⎜− ⎟ , Tcm = Tc − Tm and K cm = K c − K m jn + 1 ⎝ K m ⎠ 4. FINITE ELEMENT FORMULATION

The governing differential equation of motion for the structure is written as, [ M ]{ds } − ⎡⎣[ K e ] + P (t )[ K g ]⎤⎦ {d s } = {0}

(22)

where, [K e ] , [M] and [K g ] are the elastic stiffness, mass and geometric stiffness matrices, respectively, and { ds } and { ds } are the structural displacement and acceleration vector, respectively. In the Eq. (22), the in-plane load factor P (t ) is periodic and can be expressed in the form P (t ) = Ps + Pt cos θ t

(23)

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in which, Ps is the static portion of the load, Pt is the amplitude of the dynamic portion and

θ is the frequency of excitation. Ps and Pt are expressed in terms of static elastic buckling load Pcr of panel as, Ps = α Pcr ,

Pt = β Pcr

(24)

where α and β are static and dynamic load factors, respectively. Using Eqs. (23) and (24), the equation of motion (22) may be expressed as,

[ M ]{ds } + ⎡⎣[ K e ] − α Pcr [ K g ] − β Pcr [ K g ]cos θ t ⎤⎦ {d s } = {0}

(25)

This equation represents a system of second order differential equations with periodic coefficients of Mathieu-Hill type. The boundaries of the dynamic stability regions are formed by the periodic solutions of period T and 2T, where T = 2π / θ . The boundaries of the primary instability regions with period 2T are of practical importance and the solutions can be achieved in the form of trigonometric series as, d s (t ) =

kθ t kθ t ⎤ ⎡ { }sin { }cos a b + k k ⎢ 2 2 ⎥⎦ k =1,3,5 ⎣ ∞

∑

(26)

After substituting this in Eq. (25) and considering only the first term of the series for the regions of instability and then equating coefficients of sin

θt 2

and cos

θt 2

, Eq. (25) reduces to

⎡ ⎤ β⎞ θ2 ⎛ α [ K ] P [ K ] [ M ]⎥ {d s } = 0 ± − − ⎟ cr g ⎢ e ⎜ 2⎠ 4 ⎝ ⎣ ⎦

(27)

This equation represents an eigenvalue problem for known values of α , β and Pcr . The two conditions under plus and minus sign associated with β correspond to two boundaries of the dynamic instability region. The eigenvalues are θ , which give the boundary frequencies of the instability regions for the given values of α and β . In this analysis, the computed static buckling load of the panel is considered as the reference load. The non-dimensional excitation frequency parameter Ω = θ (a / h) 2 ρ m (1 −ν m2 ) / Em . 5. NUMERICAL RESULTS

The FG panels considered in the present studies are made of silican nitrade (ceramic) and stainless steel (metal). For these materials, the coefficients of thermal conductivity ( K c and K m ) and mass densities ( ρc and ρ m ) are assumed to be independent of temperature, and the values of the same are given by K c = 9.19 W/mK , K m = 12.04 W/mK , ρc = 2370 kg/m3

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and ρ m = 8166 kg/m3 . The elastic moduli, coefficient of thermal expansion and Poisson’s ratio are however, temperature dependent and are listed in Table 1. For the material subjected to a temperature field T, any particular elastic property P can be obtained by substituting these temperature coefficients into Eq. (10). In the present study, dynamic instability regions of cylindrical shell panels subjected to uniformly distributed in-plane load and temperature field are presented. In order to establish the correctness of the present formulation, a comparison problem of dynamic instability of FGM plates subjected to in-plane uniform load and temperature field, which was earlier solved by Lanhe et al.10 is considered and presented in Section 5.1. Table 1 Temperature dependent coefficients for silicon nitride (Si3N4) and stainless steel (SUS304) Materials

Properties

P0

Si3N4

E (Pa) η (1/K)

SUS304

E (Pa) η (1/K)

ν

ν

P−1

P1

P2 −4

9

348.43 × 10 5.8723 × 10 −6 0.24 201.04 × 109

0 0 0 0

-3.070 × 10 9.095 × 10−4 0 3.079 × 10 −4

12.33 × 10−6 0.3262

0 0

8.086 × 10−4 -2.002 × 10 −4

P3 −7

2.160 × 10 0 0 -6.534 × 10 −7 0 -3.797 × 10

-8.946 × 10−11 0 0 0

−7

0 0

5.1 Comparison problem

(i) Dynamic instability analysis of a simply supported square plate with a/h = 20, Tc = 300 K and Tm = 300 K , is carried out and the obtained results are compared with

those of Lanhe et al.10. The material properties of the constituent materials are given in Table 1. Figure 3 shows the boundaries of instability regions for two different values of static load factor α , 0.0 and 0.2, obtained by the authors along with those of Lanhe et al.10. The authors’ results are matching well with those obtained by Lanhe et al.10, which is evident from Fig. 3. The above validation studies confirm the accuracy of the present method and hence, subsequent studies are carried out using the present method. After establishing the correctness of the formulation, additional problems are considered in Section 5.2 to study the dynamic instability behaviour of FG cylindrical shell panels. 5.2 Additional problems

The geometric properties of the cylindrical panels considered in the present study are a/b = 1, a/h = 20 and unless otherwise specified, R/a = 10 and n = 1. The metal rich bottom surface is held at a temperature of 300K and the ceramic rich top surface of the panel is held at 600K. Figure 4 presents the dynamic instability regions of simply supported cylindrical FG shell panels. The values of volume fraction index are varied in order to study the effect of material 9

composition on the dynamic instability regions. Five different values of n, considered here are: 0.0, 0.5, 2.0, 10.0 and ∞. From Fig. 4, it is observed that the origin of instability shifts to lower forcing frequency and the width of the dynamic instability region decreases with the increase of the value of volume fraction index. 0.7

1 0.9 D ynam ic load factor β

Dynamic load amplitude β

0.6 0.5 0.4 Ref. [10]

0.3

Present

0.2

α=0.0

Ref. [10] 0.1 Present

0.8 0.7 0.6 0.5

n=0

0.4

n=0.5

0.3

n=2

0.2

n=10

0.1

Ceramic

α=0.2

0

0 8

12

16

20

24

28

5

32

10

20

25

30

35

40

Excitation Frequency Ω

Excitation frequency Ω

Figure 3. Comparison results of dynamic instability regions of a square simply supported FG plate

Figure 4. Dynamic instability regions for a FG cylindrical panel with Tm = 300K, Tc = 600K, R/a = 10, a/b = 1, a/h = 20 and α = 0.2 .

1.2

1.2

1

1 n = 0.0 n = 0.5 n = 5.0

0.8 0.6

α = 0.2 α = 0.4 α = 0.6

0.4 0.2

Dynamic load factor β

Dynamic load factor β

15

0.8 0.6

R/a=20

0.4

R/a=10

0.2

R/a=3

R/a=5 R/a=2

0 0

5

10

15

20

25

30

0

35

0

Excitation frequency Ω

Figure 5. Effect of static load factor on the dynamic instability regions for a FG cylindrical panel with Tm = 300K, Tc = 600K, R/a = 10, a/b = 1, a/h = 20.

10 20 30 Excitation frequency Ω

40

Figure 6. Effect of R/a ratio on the dynamic instability regions for a FG cylindrical panel with Tm = 300K, Tc = 600K, a/b = 1, n=1, a/h=20 and α = 0.2 .

Figure 5 shows the effect of static load factor α on the dynamic instability regions of simply supported FG cylindrical panels. The values of α considered in the present study are 0.2, 0.4 and 0.6. Three values of volume fraction index n, i.e. 0, 0.5 and 5 are considered. From Fig. 5, it is clearly observed that due to the increase of static component, instability regions tend to shift to lower frequencies and become wider. Here also, the effect of material composition on

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the instability regions is significant, as the instability regions shift to lower frequencies with the increase in the value of n. The effect of R/a ratio on the dynamic stability regions of simply supported FG cylindrical panels is presented in Fig. 6. It is seen from Fig. 6, that the instability region shifts to lower frequencies as the ratio of R/a increase. This is due to the fact that, with the increase in the R/a ratio, stiffness of the shell panel decreases. Figure 7 shows the effect of boundary conditions on the dynamic instability regions of FG cylindrical panels. Five different types of boundaries considered are: simply supported on all edges (SSSS), clamped on all edges (CCCC), two edges simply supported and two edges clamped (SSCC, SCSC) and corner supported, while studying their effect on the dynamic instability regions. From Fig. 7, it is observed that CCCC cylindrical panel has the widest region of dynamic instability and the origin of instability of this panel has the maximum value of excitation frequency, whereas, corner supported panel, which has the narrowest region of dynamic instability, has the minimum value of excitation frequency at the origin of instability. In the case of SSCC and SCSC panels, the instability region of SSCC panel is at the higher excitation frequency than the same for SCSC panel. 1.2

Dynamic load factor β

1 SSSS

0.8

CCCC 0.6

SSCC SCSC

0.4

Corner

0.2 0 0

10

20

30

40

Excitation frequency Ω

Figure 7 Effect of boundary conditions on the dynamic instability regions for a FG cylindrical panel with Tm = 300K, Tc = 600K, R/a = 10, a/b = 1, n=1, a/h=20 and α = 0.2

6. CONCLUSIONS

In this paper, dynamic instability behavior of functionally graded cylindrical panels is studied. The panels are in the thermal environment and are subjected to in-plane periodic loading. Several numerical problems are taken up to establish the effect of volume fraction index, static load factor, radius of curvature and boundary conditions on the instability regions. From these studies, the following concluding remarks are made.

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1. The origins and the widths of the unstable regions decrease with increasing the volume fraction index (from Fig. 4). 2. With the increase of the value of static load factor α , instability regions shift to lower frequencies and become wider (from Fig. 5). 3. The instability regions also shift to lower frequencies with the increase of the value of R/a ratio (from Fig. 6). 4. The effect of boundary conditions on the dynamic instability regions is significant. The CCCC boundary has the widest regions of instability and the corner supported FG panels has the narrowest regions (from Fig. 7). REFERENCES 1

V. Birman, “Buckling of functionally graded hybrid composite plates,” Proc. 10th Conference on Engineering Mechanics, Boulder, USA. 1995 2 C. T. Loy, K. Y. Lam and J. N. Reddy, “Vibration of functionally graded cylindrical shells,” Int. J. Solids and Struct. 41, pp. 309-324, 1999. 3 S. C. Pradhan, C. T. Loy, K. Y. Lam and J. N. Reddy, “Vibration characteristics of functionally graded cylindrical shells under various boundary conditions,” Applied Acoustics 61, pp. 111-129, 2000. 4 T. Y. Ng, K. Y. Lam and K. M. Liew, “Effects of FGM materials on the parametric resonance of plate structures,” Comput. Methods Appl. Mech. Engng. 190, pp. 953-962, 2000. 5 T. Y. Ng, K. Y. Lam, K. M. Liew and J. N. Reddy, “Dynamic stability analysis of functionally graded cylindrical shells under periodic axial loading,” Int. J. Solids and Struct. 38, pp. 1295-1309, 2001. 6 J. Yang and H. S. Shen, “Vibration characteristics and transient response of shear-deformable functionally graded plates in thermal environments,” J. Sound Vib. 255, pp. 579-602, 2002. 7 J. Yang and H. S. Shen, “Free vibration and parametric resonance of shear deformable functionally graded cylindrical panels,” J. Sound Vib. 261, pp. 871-893, 2003. 8 B. P. Patel, S. S. Gupta, M. S. Loknath and C. P. Kadu, “Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory,” Composite Structures 69, pp. 259-270, 2005. 9 J. Woo, S. A. Meguid and L. S. Ong, “Nonlinear free vibration behavior of functionally graded plates,” J. Sound Vib. 289, pp. 595-611, 2006. 10 W. Lanhe, W. Hongjun and W. Daobin, “Dynamic stability analysis of FGM plates by the moving least squares differential quadrature method,” Composite Structures 77, pp. 383-394, 2007. 11 R. Tanov, and A. Tabiei, “A simple correction to the first-order shear deformation shell finite element formulations,” Finite Elements in Analysis and Design 35, pp. 189-197, 2000.

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