analysis of functionally graded panelsdspace.nitrkl.ac.in/dspace/bitstream/2080/1101/3/ppt... ·...

1

Analysis of Functionally Graded Plates and Shells

ByDr. S. Pradyumna

Asst. Professor, Dept. of Civil Engg.,National Institute of Technology

Rourkela

ICCMS09-IIT Bombay, 1-5 Dec 2009

2

Introduction

Shell structures extensively used in aerospace, civil, marineand other engineering applications are susceptible to a varietyof time-dependent and time independent in-plane as well asout-of-plane loads.These slender structural elements under compression aremore likely to fail due to loss of stability rather than due to thestresses being higher than the strength of the material.

Structural elements subjected to in-plane periodic forces mayinduce transverse vibrations.

For certain combinations of natural frequency of transversevibration, the frequency of the in-plane forcing functions andthe magnitude of the in-plane load, resonance may occur,which is called as parametric resonance.

The spectrum of values of parameters causing unstablemotion is referred to as the regions of dynamic instability orparametric resonance

3

Background

Laminated composite structuresLight weight

Tailor made design to suit the objective

Corrosion resistance

High-strength to stiffness ratio

Mismatch of mechanical properties across an interface

Debonding at high temperatures

4

Functionally graded materials

Microscopically heterogeneous composites usuallymade from a mixture of metals and ceramics.

Mechanical properties vary smoothly from one surfaceto the other.

The gradation of properties reduces the thermalstresses, residual stresses, and stress concentrationsfound in traditional composites.

5

Applications of FGM

• Rocket heat shields • Heat exchanger tubes • Thermoelectric generators • Heat-engine components • Wear-resistant linings for handling heavy

abrasive ore particles• Electrically insulating metal/ceramic joints

FGM

Ceramic High temperature resistance due to its low thermal conductivity

Metal Structural strength and fracture toughness.

6

Shell theories

Love-Kirchoff’s classical theory

First-order shear deformation theory (FSDT)

Higher-order shear deformation theory (HSDT)

7

Literature review- Higher-order theories

Yang [1973] Included all three radii of curvature (Rx , Ryand Rxy). The displacement functions for u, v and w areexpressed as products of one-dimensional, first-orderHermite interpolation formulas.

The strain displacement relations are given by,

' / ,u w Rx xε = − / ,oy yv w Rε = − ' 2 /o

xy xyv u w Rγ = + −'' ' '/ / ,x x xyw u R v Rκ = − − − '' / /o o

y y xyw v R u Rκ = − − −

' ' '2 / / ( )/o o oxy x y xyw u R v R u v Rκ =− − − − +

8

Bhimaraddi [1984]-Higher-order theory for free vibration analysis of circular cylindrical shells

Reddy and Liu [1985] -Modified the Sanders’ theory

Parabolic distribution of the transverse shearstrain through thickness

Tangential stress-free boundary conditions onthe boundary surfaces

The displacement field is assumed as,

32

30 2

0

0

43

413

1 y

x xx

yy

wzh y

z wu u z zR h x

zv v zR

w w

θ

θ θ

θ

⎛ ⎞ ⎡ ⎤⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠⎛ ⎞ ⎡ ⎤∂

− +⎜ ⎟ ⎢ ⎥⎜ ⎟ ∂⎣ ⎦⎝ ⎠

∂= + + − +∂

= + +

=

9

Kant and Menon [1989] Developed theories for both geometrically thin and thick shells.

Mallikarjuna and Kant [1992] Refined shell theory for the analysis of laminated and sandwich shells.

Liew and Lim [1996] Higher-order theory by considering the Lamé parameters and for the transverse shear strains.

Kant and Khare [1997] Presented a higher-order facet quadrilateral composite shell element.

(1 / )yz R+(1 / )xz R+

10

Functionally graded panelsFuchiyama and Noda (1995)- Analyzed transient heat transfer

and transient thermal stress of FGM by the FEM.Aboudi et al. (1999) Developed a Cartesian co-ordinate based

higher-order theory for FGMsReddy (2000) Navier’s solutions and finite element results

based on the TSDT for FG plates.

Buckling analysisShahsiah and Eslami (2003)- Thermal buckling of FGcylindrical shells Kadoli and Ganesan (2006)-Thermal buckling and free vibration analyses of clamped-clamped FG cylindrical shellsGanapathi and Prakash (2006) and Ganapathi et al. (2006)-Buckling of simply supported FG skew plates subjected to thermal and mechanical loads, respectively.

11

Dynamic instability analysis

Ng et al. (2000, 2001), Yang and Shen (2003)- Dynamic stability of FG plates and CYL shells

Lanhe et al. (2007)- Dynamic instability of FG plates using MLSDQ

Ganapathi (2007)- Dynamic instability of FG spherical shells

12

Scope and Objectives

Scope of the present investigation is primarily toanalyze different shell panels employing a higher‐order shear deformation theory (HSDT)

Buckling and dynamic instability analyses of FG shellpanels‐cylindrical (CYL), spherical (SPH) and hypar(HYP)

The material properties of the FG shell panels are tobe taken thermal dependent and heat conduction isconsidered.

13

Mathematical Formulation

The displacement components u(x,y,z), v(x,y,z) and w(x,y,z) at any point in the plate space are expanded in a Taylor’s series in terms of the thickness coordinates.

The displacement field may be assumed in the form [Kant and Khare -1997]

2 * 3 *0 0

2 * 3 *0 0

0

( , , ) ( , ) ( , ) ( , )

( , , ) ( , ) ( , ) ( , )( , , )

y y

x x

u x y z u x y z z u x y z x y

v x y z v x y z z v x y z x yw x y z w

θ θ

θ θ

= + + +

= − + −=

14

The strain-displacement relations are,

Strain components

xx

u wx R

ε ∂= +∂

yy

v wy R

ε ∂= +∂

2x y

x y

v u wx y R

γ ∂ ∂= + +

∂ ∂

1 1xzx xy

u w u vC Cz x R R

γ ∂ ∂= + − −∂ ∂ 1 1yz

y xy

v w v uC Cz y R R

γ ∂ ∂= + − −∂ ∂

2 * 3 *0 0

2 * 3 *0 0

2 * 3 *0

2 * 3 *

2 * 3 *

x x x x x

y y y y y

x y x y x y x y x y

x z x x z x x z

y z y y z y y z

z z z

z z z

z z z

z z z

z z z

ε ε κ ε κ

ε ε κ ε κ

γ γ κ γ κ

γ ϕ κ ϕ κ

γ ϕ κ ϕ κ

= + + +

= + + +

= + + +

= + + +

= + + +

15

Where,

{ }* * * *

0 0 0 0 0 0 0 0 0 0 0* * *0 0 0 0 0 0

2, , , , ,, , , , ,x y xy x y xyx y xy

u w v w u v w u v v ux R y R y x R x y x y

ε ε γ ε ε γ⎧ ⎫∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪+ + + + += ⎨ ⎬∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪⎩ ⎭

{ }* ** *

* * * 0 00 0, , , , , , , , , ,y y y yx x x x

x y xy x y xyu vC C

x y y x y x x y y xθ θ θ θθ θ θ θκ κ κ κ κ κ

⎧ ⎫∂ ∂ ∂ ∂⎪ ⎪∂ ∂ ∂ ∂ ∂ ∂= − − + − − −⎨ ⎬∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎪ ⎪⎩ ⎭

{ } 0 0 0 0 01 1 1 1

1

,, y xx yx xy y xy

w u v v uwC C C Cx R R y R R

θ θϕ ϕ∂ ∂⎧ ⎫+ − − − − −= ⎨ ⎬∂ ∂⎩ ⎭

{ }* * * *

* *0 0 0 0* *1 1 1 13 , 3, y xx y

x x y y x y

u v v uC C C CR R R R

θ θϕ ϕ⎧ ⎫⎪ ⎪− − − − −= ⎨ ⎬⎪ ⎪⎩ ⎭

{ }* ** *

* ** *1 1 1 1 1 1 1 12 ,2 , ,, , , y y y yx x x x

o oxz yz xz yzx xy y xy x xy x xy

u C C v C C C C C CR R R R R R R Rθ θ θ θθ θ θ θ

κ κ κ κ⎧ ⎫⎪ ⎪− + + − − + −= ⎨ ⎬⎪ ⎪⎩ ⎭

1C

C1 is a tracer by which the analysis can be reduced to that of shear deformable Love’s first approximation and is the result of Sander’s theory which accounts for the condition of zero strain for rigid body motion.

( )0 0.5 1/ -1/x yC R R=

16

Functionally graded materialsEffective material property P (such as Young’smodulus, Poisson’s ratio, mass density etc.) is givenby

is the volume fraction of ceramic andis the volume fraction of the metaland are related by

c c m mP PV P V= +

1c mV V+ =

22

n

ch zV

h+⎛ ⎞= ⎜ ⎟

⎝ ⎠and , where n is the material variation

profile through the panel thickness

17

Material properties

18

Material properties

19

Stress-strain relationship

….(1)

11 12

12 22

66

44

55

0 0 0 10 0 0 1

0 0 0 0 ( ) ( )00 0 0 0 00 0 0 0 0

x x

y y

xy xy

yz yz

xz xz

Q QQ Q

Q z T zQ

Q

σ εσ ετ γ ατ γτ γ

⎛ ⎞⎧ ⎫ ⎡ ⎤ ⎧ ⎫ ⎧ ⎫⎜ ⎟⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎜ ⎟⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎜ ⎟⎢ ⎥= − Δ⎨ ⎬ ⎨ ⎬ ⎨ ⎬⎜ ⎟⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎜ ⎟⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎜ ⎟⎢ ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭⎩ ⎭ ⎣ ⎦ ⎩ ⎭⎝ ⎠

In which, 11 22 2

( )1 ( )

E zQ Qzν

= =− 12 2

( ) ( )1 ( )

z E zQz

νν

=− 44 55 66

( )( )2(1 ( ))

E zQ Q Q G zzν

= = = =+and

{ } [ ]{ }Qσ ε=Eq. (1) is re-written as

[ ]Qwhere is effective stiffness coefficient matrix given by the relation, 2[ ] ( )

2

n

c m mz hQ Q Q Q

h+⎛ ⎞= − +⎜ ⎟

⎝ ⎠

In which, Qc and Qm are effective stiffness coefficient matrices for ceramic and metal constituents, respectively.

20

*/ 2

* * 2

* / 2

*/ 2

* * 3

* / 2

* ** *

* */

, 1,

, ,

[ , , , ]

x x xh

y y yh

xy xy xy

x x xh

y y yh

xy xy xy

xzx x x x

yzy y y y h

N NN N N N z dz

N N

M MM M M M z z dz

M M

Q S Q SQ S Q S

Q S Q S

σστ

σστ

ττ

−

−

−

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

∫

∫

/ 22 3

2

1, , ,h

z z z⎡ ⎤⎣ ⎦∫

or , whereDσ ε=( )* * * * * * * * * *, , , , , , , , , , , , , , , , , , ,

T

x y xy x y xy x y xy x y xy x y x y x y x yN N N N N N M M M M M M Q Q Q Q S S S Sσ =

Integrating the stresses through the thickness, the resultant forcesand moments acting on the panel are obtained.

21

[ ]00

0 0

m cTc b

s

D DD D D

D

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

in which,

The elements of Dc and Db can be obtained by replacing H1 ,H3 and H5by (H2 ,H4 and H6) and (H3,H5and H7), respectively and

( )* * * * * * * * * *0 0 0 0 0 0, , , , , , , , , , , , , , , , , , ,

T

x y xy x y xy x y xy x y xy x y x y xz yz xz yzε ε ε γ ε ε γ κ κ κ κ κ κ ϕ ϕ ϕ ϕ κ κ κ κ=

⎡ ⎤= ⎢ ⎥⎣ ⎦

1 3m

3 5

H HD

H H

( )/ 2

2 3 4 5 6

/ 2

( ) 1, , , , , , h

ijh

z z z z z z dz−

= ∫1 2 3 4 5 6 7H ,H ,H ,H ,H ,H ,H Q

sym

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

1 3 2 4

5 4 6s

3 5

7

H H H HH H H

DH H

H

22

Thermal stress resultants/ 2

/ 2

{ } ( ) h

T Tx y

h

N N T z dzβ−

= = Δ∫/ 2

/ 2

{ } ( ) h

T Tx y

h

M M T z z dzβ−

= = Δ∫

In which

{ }11 12

12 22

( ) ( )

( ) ( )0

Q Q z

Q Q z

α

β α

⎧ ⎫+⎪ ⎪⎪ ⎪= +⎨ ⎬⎪ ⎪⎪ ⎪⎩ ⎭

For a functionally graded structure, the temperature change throughthickness is not uniform, and is governed by the one-dimensionalFourier equation of heat conduction.

d d( ) 0d d

TK zz z⎡ ⎤− =⎢ ⎥⎣ ⎦

----- (2)

subjected to conditions, (at / 2) and (at / 2)= = = = −c mT T z h T T z hThe solutions of Eq. (2) can be obtained by means of polynomial series

23

15

0

1 2( )1 2

j jncm cm

mj m

T K z hT z Tjn K hξ

+

=

⎛ ⎞ +⎛ ⎞= + −⎜ ⎟ ⎜ ⎟+ ⎝ ⎠⎝ ⎠∑

5

0

11

j

c m

j m

Kjn K

ξ=

⎛ ⎞= −⎜ ⎟+ ⎝ ⎠∑

cm c mT T T= −

cm c mK K K= −

Energy expressions

Strain energy due to linear strains

due to non-linear strains

Kinetic energy

1 { } { }2

T

l lV

U dvε σ= ∫1 { } { }2

T

nl nlV

U dvσ ε= ∫

2 2 21 ( )2 V

T u v w dvρ= + +∫ & & &

Governing differential equation

According to the extended Hamilton’s principle, the governing

Equation is given as: 2

1

( ( ) ) 0t

l nlt

T U P t U dtδ − − =∫ ----- (3)

24[ ][ ]{ } [ ] ( )[ ] { } { }e GM d K P t K d Q+ − =&&

2

1

1 1 1{ } [ ]{ } { } [ ]{ } ( ){ } [ ]{ } { } [ ] { } d d 02 2 2

tTT T T T

e gt A

d m d d k d P t d k d d N p A tδ⎡ ⎤

− − − =⎢ ⎥⎣ ⎦

∫ ∫& &

Substituting the corresponding terms in Eq. (3),

2

1

{ } [ ]{ } [ ]{ } ( )[ ]{ } [ ] { } d d 0t

T Te g

t A

d m d k d P t k d N p A tδ⎡ ⎤

− − − =⎢ ⎥⎣ ⎦

∫ ∫&&or

[ ]{ } [ ]{ } ( )[ ]{ } [ ] { } dTe g

A

m d k d P t k d N p A⎡ ⎤− − =⎣ ⎦ ∫&&

[ ]{ } [ ] ( )[ ] { } { }e gm d k P t k d q⎡ ⎤− + =⎣ ⎦&&

{ } [ ] { } dT

A

q N p A= ∫in which is the element load vector

After assembling the element matrices, the governing differential equation becomes

----- (4)

25

Governing Equations

Static analysis

Free vibration analysisSetting P(t) = 0 and {Q} = 0 in Eq. (4)Here, the displacement is a function of space and time. Forsolving the free vibration problem, separation of space andtime co-ordinates is done by the following substitution

By differentiating two times with respect to time:

[ ]{ } { }e sK d Q=

[ ]{ } [ ]{ } 0s e sM d K d− =&&

{ ( , , , )} { ( , )}i tsd x y z t ae x yω φ= { } { }i t

sd ae ω φ=or

2{ } { }i tsd a e ωω φ= −&&

----- (6)

----- (7)

Combining Eqs. (6) and (7) in Eq. (5) and assuming that

the solution for free vibration equation is achieved in the form of

- (5)

0i tae ω ≠

26

{ }2[ ] [ ] 0K Mω φ⎡ ⎤− =⎣ ⎦ ----- (8)

where and represent the natural frequencies (eigenvalues)and the corresponding mode shapes (eigenvectors) of thegeneralized eigenvalue problem.

The lowest value of natural frequency is termed as thefundamental natural frequency of the structure.

ω { }φ

Buckling problem

Considering only static part of the in-plane load in ,

and acceleration Eq. (4) becomes the governing equation for a static buckling problem with in-plane load.

Eigenvalues of the above governing equation are the buckling loads for different modes.

( )( )P t P=

{ } 0Q = { } 0sd =&&

[ ] [ ] { } 0e cr g sK P K d⎡ ⎤+ =⎣ ⎦----- (9)

27

The in-plane load in Eq. (4) is periodic ( ) coss tP t P P t= + Ω

s crP Pα= t crP Pβ=

Static load factor Dynamic load factor

[ ]{ } [ ] [ ] [ ]cos { } { }α β⎡ ⎤+ − − Ω =⎣ ⎦&&

s e cr g cr g sM d K P K P K t d Q

This equation represents a system of second order differentialequations with periodic coefficients of Mathieu-Hill type. Thedevelopment of dynamic instability regions arises from Floquet’stheory, which establishes the existence of periodic solutions. Theboundaries of the dynamic stability regions are formed by theperiodic solutions of period T and 2T, where .2 /π= ΩT

----- (10)

Dynamic instability problem

28

1,3,5( ) { }sin { }cos

2 2s k kk

k t k td t a b∞

=

Ω Ω⎡ ⎤= +⎢ ⎥⎣ ⎦∑

Substituting the above equation in Eq. (10) and considering only the first term of the series for the regions of instability and then equating coefficients of and , Eq.(10) reduces to,

s in2Ω t cos

2Ωt

2

[ ] [ ] [ ] [ ] { } 02 4e cr G cr G sK P K P K M dβα

⎡ ⎤Ω− ± − =⎢ ⎥

⎣ ⎦----- (11)

The boundaries of the primary instability regions with period 2Tare of practical importance and the solutions can be achieved in the form of trigonometric series as,

29

x

yBoundary conditions

Simply Supported (SS) * *0 0 0 0,y yv w vθ θ= = = = = at x = 0, a;

* *0 0 0 0,x xu w uθ θ= = = = = at y = 0, b

a

b

Clamped (CL) * * * *0 0 0 0 0 0,x y x yu v w u vθ θ θ θ= = = = = = = = = at x = 0, a; y = 0, b

Corner Supported (SS) All degrees of freedom at the corners are restrained

30

Results and Discussion

shell panels considered

z

x c

a

b

4cz xyab

=

c c

c

y

31

Material properties

Table 1 Temperature dependent coefficients for Silicon nitride (Si3N4) and Stainless steel (SUS304) (Huang and Shen, 2004)

Material (M1) Properties 0P 1P− 1P 2P 3P

Si3N4 Ec (Pa) 9348.43 10× 0 4-3.070 10−× 72.160 10−× 11-8.946 10−×

cα (1/K) 65.8723 10−× 0 49.095 10−× 0 0

cν 0.24 0 0 0 0

cρ (kg/m3) 2370 0 0 0 0

K (W/mK) 9.19 0 0 0 0 SUS304 Em (Pa) 9201.04×10 0 43.079 10−× 7-6.534 10−× 0

mα (1/K) 612.33 10−× 0 48.086 10−× 0 0

mν 0.3262 0 4-2.002 10−× 7-3.797 10−× 0

mρ (kg/m3) 8166 0 0 0 0

K (W/mK) 12.04 0 0 0 0

Table 2 Material properties of Aluminum-Alumina (Praveen and Reddy, 1998)

Material (M2) Young’s modulus (GPa)

Poisson’s ratio Mass density (kg/m3)

Aluminum (metal) 70mE = 0.3mν = 2707mρ =

Alumina (ceramic)

380cE = 0.3cν = 3000cρ =

32

Table 3 Non-dimensional parameters of FG shell panels

Parameter Non-dimensional parameter

Natural frequency (λ ) 2 */ω ρm ma h D

Buckling load ( )γ ( )2 2cr / π cP b D

Excitation frequencies ( )Ω 2 */ρΩ m ma h D

* 3 2/12(1 )m m mD E h ν= − ; 3 2/12(1 )ν= −c c cD E h

Non-dimensional parameters

33

Table 4 Non-dimensional frequency parameters for Si3N4/SUS304 square plates in thermal fields Temperature field n FSDT Present

HSDT Ref. A

Tm = 300 K, Tc = 300 K

0.0 12.3926 12.1382 12.495 0.5 8.5545 8.4055 8.675 1.0 7.5091 7.3992 7.555 2.0 6.7402 6.6506 6.777 ∞ 5.3609 5.4341 5.405

Tm = 300 K, Tc = 400 K

0.0 12.2654 12.0172 12.397 0.5 8.4973 8.3495 8.615 1.0 7.4713 7.3610 7.474 2.0 6.7148 6.6245 6.693 ∞ 5.3609 5.4322 5.311

Tm = 300 K, Tc = 600 K

0.0 11.7966 11.8126 11.984 0.5 8.2691 8.2554 8.269 1.0 7.3057 7.2969 7.171 2.0 6.5900 6.5807 6.398 ∞ 5.3471 5.4296 4.971

Free vibration analysis- Comparison studies

Pradyumna and Bandyopadhyay, J. Engg. Mech. ASCE, (In press)

34

Table 5 Comparison of non-dimensional frequency parameter 2 /m mh Eλ ω ρ= for a simply supported Aluminum-Zirconia plate

Results n = 1 a/h = 5 n = 0

a/h = 5 a/h = 10 a/h = 20 n = 2 n = 3 n = 5 a/h = 10 a/h =10Present HSDT 0.2257 0.0613 0.0157 0.2237 0.2243 0.2253 0.4658 0.0578

FSDT 0.2323 0.0633 0.0162 0.2325 0.2334 0.2334 0.4619 0.0577

3-D (Ref. A) 0.2192 0.0596 0.0153 0.2197 0.2211 0.2225 0.4658 (0.5535*)

0.0578 (0.0592*)

HSDT (Ref. B) 0.2285 0.0618 0.0158 0.2264 0.227 0.2281 0.4658 0.0578 Ref. C 0.2188 0.0592 0.0147 0.2153 0.2202 0.2215 - - Ref. D 0.2188 0.0584 0.0149 0.2153 0.2172 0.2194 - -


For Aluminum: 70 GPa=mE , 0.3ν =m , 32702 kg/mmρ = and for Zirconia:

200 GPa=cE , 0.3ν =c , 35700 kg/mcρ = .

Material properties

Ref. A – Vel and Batra (2004); Ref. B- Matsunaga (2008);

Ref. C- Ferreira et al. (2006); Ref. D- Qian et al. (2004).

35

Table 6 Comparison of non-dimensional frequency parameter λ for a clamped FG cylindrical shell panel

Mode Material Composition Si3N4 n = 0.2 n = 2.0 n = 10.0 SUS304

1 Present HSDT 73.2821 59.3633 39.9193 34.5657 32.5474 Ref. A 74.518 57.479 40.750 35.852 32.761




Ref. A – Yang and Shen (2003)


a = b, a/h = 10, a/Rx = 0.1 and Tm = Tc = 300 K

Pradyumna and Bandyopadhyay, J. Sound. Vib., 318, 176-192, 2008

0.0 0.2 0.4 0.6 0.8-3

-2

-1

0

1

2

T c = 400 K T m = 300 K

Cen

tral d

ispl

acem

ent (

mm

)

T im e (m s)

Ref. A n = 0.5 Present n = 0.5 Ref. A n = 2.0 Present n = 2.0

Dynamic response

Transient response of simply supported FGM platewith different values of n Ref. A – Huang and Shen (2004)

0 .0 0 .2 0 .4 0 .6 0 .8-3

-2

-1

0

1

2

3

Cen

tral d

ispl

acem

ent (

mm

)

T im e (m s)

T c = 300K T m = 300K P resen t T c = 600K T m = 300K P resen t T c = 300K T m = 300K R e f. A T c = 600K T m = 300K R e f. A

Transient response of simply supported FGMplate with different values of Tc

Ref. A – Huang and Shen (2004)

38

Buckling of FG shell panels- Comparison study

Table 7 Comparison of buckling load parameter ( 2 2/iso crP b Dγ π= )

for CL isotropic rectangular plate ( 0.3ν = )

a/h a/b HSDT FSDT Liew et al. (2003b)-HSDT

Wang et al. (1993)- FSDT

10 1 8.3025 8.0551 8.3329 8.2733 0.5 13.1821 12.2383 13.2287 12.974

20 1 9.3521 9.5139 9.3884 9.5526

0.5 17.1356 16.8012 17.1636 17.199

Pradyumna and Bandyopadhyay, IJSSD, 10, 4, 2010 (In press)

39

Buckling of FG shell panels- Comparison study

Table 8 Comparison of critical load parameter γ for a CL plate made up of Aluminum-Zirconia

a/h = 20 a/h = 40 PresentHSDT

PresentFSDT Ref. B Present

HSDT PresentFSDT Ref. B

n = 0.0 9.2760 9.5190 9.3922 9.7985 10.04379.6938n = 2.0 5.9325 6.0970 6.0544 6.2757 6.4355 6.2517n = 5.0 5.4752 5.6365 5.6770 5.8111 5.9627 5.8829n = ∞ 4.5892 4.4128 4.3540 4.8573 4.6560 4.4938

Ref B. Liew et al. (2003)

For Aluminum: 70 GPa=mE , 0.3ν =m , 32702 kg/mmρ = and for Zirconia:

200 GPa=cE , 0.3ν =c , 35700 kg/mcρ = .

Material properties

40

Table 9 Non-dimensional buckling loads of FGM shell panels with SS boundary conditions

n CYL a/Rx 0 1/20 1/10 1/5 1/3

0 7.4713 7.6130 7.8360 8.3737 8.6193 1 7.3338 7.3812 7.5010 7.6998 7.9168

10 7.3352 7.3922 7.4721 7.6701 7.9287 ∞ 7.2721 7.3608 7.4698 7.7132 7.9986

n SPH a/Rx 0 1/20 1/10 1/5 1/3

0 7.4713 7.8453 8.4649 9.1729 10.1485 1 7.3338 7.5131 7.7841 8.3776 9.2483

10 7.3352 7.4919 7.7537 8.3703 9.1506 ∞ 7.2721 7.4939 7.8040 8.4501 9.1952

n HYP c/a 0 0.05 0.1 0.15 0.2

0 7.4713 8.1125 8.1732 8.2647 8.3541 1 7.3338 7.5406 7.6060 7.6875 7.7611

10 7.3352 7.3489 7.3887 7.4453 7.4912 ∞ 7.2721 7.2492 7.2686 7.3191 7.3620

41

Dynamic Instability Analysis of FG shell panels

Comparison of dynamic instability regions of FG plate, a/b = 1, a/h = 20, SS, Tm = 300 K, Tc = 300 K, α = 0.2. Ref. A- Lanhe et al. (2007)

42

a/b = 1, a/h = 20, Tm = 300 K, Tc = 600 K, α = 0.2, SS, n = 1.0

Effect of curvature on the dynamic instability regions of

FG-CYL shell panel


FG-SPH shell panel

20 30 40 50 60 70 800.0

0.2

0.4

0.6

0.8

1.0

1.2CYL

β

Ω

a/Rx = 1/3 a/Rx = 1/5 a/Rx = 1/10 a/Rx = 1/20 a/Rx = 1/50

20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

1.2SPH

β

Ω

a/Rx = 1/3 a/Rx = 1/5 a/R

x = 1/10

a/Rx = 1/20 a/Rx = 1/50

43

0 40 80 120 1600.0

0.2

0.4

0.6

0.8

1.0

1.2HPR

β

Ω

c/a = 0.00 c/a = 0.05 c/a = 0.10 c/a = 0.15 c/a = 0.20


FG-HPR shell panel

Pradyumna and Bandyopadhyay, J. Engg. Mech. ASCE, (In press)

44

Effect of material composition on the dynamic instability regions of

FG shell panels

CYL

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120

Ω

β

n = 0.0n = 0.5n = 1.0n = 10.0n = ∞

Effect of volume fraction index on dynamic instability regions of FG-shell panels; a/b = 1, a/h = 20, SS, Tm = 300 K, Tc = 600 K, a = 0.2, a/Rx = 10 (for CYL and SPH), c/a = 0.2 (for hypar)

HYP

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150 200 250

Ω

βn = 0.0n = 0.5n = 1.0n = 10.0n = ∞

45

Effect of static load factor on the dynamic instability regions of

FG shell panels

SPH

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80

Ω

β

α = 0.0α = 0.2α = 0.4α = 0.5α = 0.6

HYP

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 50 100 150

Ω

β

α = 0.0α = 0.2α = 0.4α = 0.5α = 0.6

Effect of volume fraction index on dynamic instability regions of FG-shell panels; a/b = 1, a/h = 20, SS, n =1; Tm = 300 K, Tc = 600 K, a/Rx = 10 (for CYL and SPH), c/a = 0.2 (for hypar)

46

Effect of boundary conditions on the dynamic instability regions of FG shell panels

47

Buckling loads increase with the increase of curvatureparameter,

Buckling load decreases with the increase of volumefraction index n, i.e., when the shell panels are metal richfor which the corresponding E is reduced.FG shell panels subjected to uniformly distributed uni-axial load and temperature field, buckle at higher loadwith the increase in number of edge constraints.Accordingly, CL shell panels give higher values offollowed by SS and CS shell panels.

Conclusions

48

For all shell panels, the origin of instability region shiftsto lower excitation frequency and becomes narrow withthe increase of volume fraction index n. This seems to bedue to decreasing of both mass and stiffness.

FG shell panels with CL boundary have the widestregions of dynamic instability and the origin of instabilityhas the maximum value of excitation frequency. This isfollowed by shell panels with SS and CS boundaryconditions.

The instability region tends to shift to lower frequenciesand become wider with the increase of static loadcomponent α.

Conclusions (Contd..)

49

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References

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