Mechanical and Thermal Buckling of Functionally Graded Plates Roman - - PowerPoint PPT Presentation

Mechanical and Thermal Buckling

• f Functionally Graded Plates

Roman Arciniega and J. N. Reddy

Department of Mechanical Engineering Texas A&M University College Station, TX 77843-3123, USA

US-South American Workshop: Mechanics and Advanced Materials − Research and Education August 2-6, 2004

CONTENTS OF THE LECTURE

• FE Model
• Theoretical Formulation
• Background
• Numerical Results
• Closing comments

Functionally graded materials are inhomogeneous

materials in which the material properties are varied continuously from point to point to eliminate interface problems and thus the stress distributions are smooth and uniform.

For example, a plate structure used as a thermal

barrier may be graded through the plate thickness from ceramic on the face of the plate that is exposed to high temperature to metal on the other face.

BACKGROUND BACKGROUND

FGM MATERIALS FGM MATERIALS

P(z) = (Pc − Pm) f(z) + Pm f(z) = [(2z + h)/2h)]n Pc (ceramic) Pm (metal)

x z

y x z h

Pc (ceramic)

ceramic metal ceramic metal metal

This is achieved by varying the volume fraction

• f the constituents i.e., ceramic and metal in a

predetermined manner. The ceramic constituent

• f the material provides the high temperature

resistance due to its low thermal conductivity. The ductile metal constituent, on the other hand, prevents fracture caused by stresses due to high temperature gradient in a very short period of time.

Through Through-

• thickness

thickness-

• Graded

Graded Material Plates Material Plates

A Typical Metal A Typical Metal− −Ceramic FGM Ceramic FGM

High temperature Ceramic Heat resistant; side good anti-oxidant property; low thermal conductivity Low temperature Metal Mechanical strength; side high thermal conductivity; high fracture toughness In between Ceramic Effective thermal stress & metal relaxation throughout

Pc (ceramic) Pm (metal)

x z

Aluminum and Aluminum and Zirconia Zirconia Material Properties Material Properties

E1 = 70 GPa, ν = 0.3, ρ = 2707 kg/m3, k = 204 W/(m.K), α = 23 × 10-6 / oC Aluminum Zirconia E1 = 151 GPa, ν = 0.3, ρ = 3000 kg/m3, k = 2.09 W/(m.K), α = 23 × 10-5 / oC

) ( = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − dz dT z k dz d

Center deflection:

h w w =

Load parameter:

4 4

h E a q P

m

• =

The temperature distribution over the thickness is determined by solving the energy equation Non-dimensional parameters

Temperature variation through the Temperature variation through the aluminum aluminum-

• zirconia

zirconia FGM plate thickness FGM plate thickness

SUMMARY

The mechanical and thermal buckling of functionally graded ceramic-metal plates is investigated. The third-order shear deformation theory for plates is used. A displacement finite element model of the third-order theory is developed using c0-continuity with a family of high-order Lagrange interpolation functions to avoid shear locking. The stability equations are derived using the Trefftz criterion. Numerical results are compared with those of the FSDT.

RELATED RESEARCH

• Most of research studies in FGMs had more focused on

thermal stress analysis and fracture mechanics (Paulino and his colleagues). Limited work has been done to study the buckling and vibration response of FGM structures. Javaheri and Eslami derived the stability equations of FGM plates under thermal loads, based on the CLPT. Shen carried out a postbuckling analysis for FGM panels and plates subjected to axial compression in thermal environments. Na and Kim presented a 3D finite element solution for thermal buckling of functionally graded plates.

THEORETICAL FORMULATION

KINEMATICS

w z y x u w k z v z y x u w k z u z y x u

y x

= + + + = + + + = ) , , ( ) , ( ) , , ( ) , ( ) , , (

3 2 2 2 1 1 1

ϕ ϕ ϕ ϕ

Displacement Field of TSDT

x,u1 y,u2 a b h z,u3 x,u z,w h/2 h/2 Ceramic Metal

To relax the continuity in the finite element formulation, we introduce the following auxiliary variables

   + = + =

2 2 1 1

, , ϕ ψ ϕ ψ

y x

w w

w z y x u k z v z y x u k z u z y x u = + + = + + = ) , , ( ) , , ( ) , , (

3 2 2 2 1 1 1

ψ ϕ ψ ϕ

The strains are

2 2 3 3 1

z k z k z k

m

• m

m i i

• i

i

+ = + + = ε ε ε ε

) 5 , 4 ( ) 6 , 2 , 1 ( = = m i

KINEMATICS (continued)

where

1 , 5 2 , 4 , , , , 6 2 , , 2 2 , , 1

, , , 2 1 , 2 1 ϕ ε ϕ ε ε ε ε + = + = + + = + = + =

x

• y
• y

x x y

• y

y

• x

x

• w

w w w v u w v w u

x y x y y x x y y x

k k k k k k k k k k k k k

, 1 2 5 , 2 2 4 , 2 , 1 3 6 , 2 3 2 , 1 3 1 , 2 , 1 1 6 , 2 1 2 , 1 1 1

3 , 3 ), ( , , , , , ψ ψ ψ ψ ψ ψ ϕ ϕ ϕ ϕ = = + = = = + = = = and

KINEMATICS (continued)

EQUILIBRIUM EQUATIONS

∂ ∂ ∂ ∂ : ∂ ∂ ∂ ∂ : ∂ ∂ ∂ ∂ :

2 1 2 6 6 1

= + + + = + = + N q y Q x Q w y N x N v y N x N u δ δ δ 3

• ∂

∂ ∂ ∂ : 3

• ∂

∂ ∂ ∂ :

• ∂

∂ ∂ ∂ :

• ∂

∂ ∂ ∂ :

2 2 6 2 1 6 1 1 2 2 6 2 1 6 1 1

= + = + = + = + R y P x P R y P x P Q y M x M Q y M x M δψ δψ δϕ δϕ

∫ ∫ ∫

− − −

= = = =

2 2 2 4 2 2 2 2 2 5 1 1 2 2 3

) , 1 ( ) , ( , ) , 1 ( ) , ( ), 6 , 2 , 1 ( ) , , 1 ( ) , , (

h h h h h h i i i i

dz z R Q dz z R Q i dz z z P M N σ σ σ

MECHANICAL CHARACTERIZATION OF FUNCTIONALLY GRADED PLATES Volume Fractions:

m c cm m m c c

E f E f E f E z E + = + = ) (

m c cm m m c c

f f f z α α α α α + = + = ) (

ν ν = ) (z

c m n c

f f h z f − =       + = 1 2 1

Constant Poisson’s coefficient

CONSTITUTIVE EQUATIONS:

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧

5 4 44 5 4 6 2 1 66 22 12 12 11 6 2 1

) ( 55 ) ( ) ( ) ( ) ( ) ( ) ( ε ε σ σ ε ε ε σ σ σ z Q z Q z Q z Q z Q z Q z Q

) 1 ( 2 ) ( ) ( ) ( ) ( , 1 ) ( ) ( , 1 ) ( ) ( ) (

55 44 66 2 12 2 22 11

ν ν ν ν + = = = − = − = = z E z Q z Q z Q z E z Q z E z Q z Q

WHERE:

STRESS RESULTANTS:

3 1 3 1 3 1 j ij j ij j ij i j ij j ij j ij i j ij j ij j ij i

k H k F E P k F k D B M k E k B A N + + = + + = + + = ε ε ε

) 6 , 2 , 1 , ( = j i

2 4 4 2 2 5 5 1 2 4 4 2 2 5 5 1

, ,

j j j j j j j j j j j j j j j j

k F D R k F D R k D A Q k D A Q + = + = + = + = ε ε ε ε

) 5 , 4 , ( = j i

THE MATERIAL STIFFNESS COEFFICIENTS FOR FGMs:

∫ ∫

− −

+ = + =

2 2 4 2 2 2 6 4 3 2

) , , 1 )( ( ) , , , ( ) , , , , , 1 )( ( ) , , , , , (

h h m ij c cm ij ij ij ij h h m ij c cm ij ij ij ij ij ij ij

dz z z Q f Q F D A dz z z z z z Q f Q H F E D B A

RECALL:

m ij c ij cm ij

Q Q Q − =

THE THERMAL STRESSES ARE:

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

66 22 12 12 11 6 2 1

z y x T z z z Q z Q z Q z Q z Q

T

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ α α σ σ σ

THERMAL STRESS RESULTANTS:

∫ −

+ + − − =

2 2 2)

, , 1 ( ) , , ( ) )( ( ) 1 ( 1 ) , , (

h h m c cm m c cm T T T

dz z z z y x T f E f E P M N α α ν

STABILITY ANALYSIS

The potential energy increment may be written in the form

∑

=

Π = Π − + Π = Π

4 1

1

n n F I F

n δ δ ! ) U ( ) U U (

: Fundamental prebuckling solution

I

U

F

U

: Incremental solution

Necessary condition

= Π δ

The critical load is defined as the smallest load for which the second variation is no longer positive definite. The limit of positive-definiteness for a continuous system can be expressed as

] [

2

= Π δ δ

The fundamental solution (prebuckling state) is considered as a pure membrane state where bending and rotations are

• neglected. Then

⎪ ⎪ ⎩ ⎪ ⎪ ⎨ ⎧ = = = = = + = + = →

1 2 2 1 1 1 1 2 2 1 1 1 1 1 1

, , , , , , ψ ψ ψ ψ ϕ ϕ ϕ ϕ w w v v v u u u U

(THE TREFFTZ CRITERION)

STABILITY ANALYSIS (continued)

) ( ) (

2 2

= Π + Π

FI I

δ δ δ δ

Incremental Fundamental-incremental

∫

Ω

+ + + = Π dxdy w w N w w w w N w w N

y y x y y x x x FI

} {

, 1 , 1 2 , 1 , 1 , 1 , 1 6 , 1 , 1 1 2

) ( ) ( δ δ δ δ δ δ

where

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ∂ ∂ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ ∂ ∂ = y w N x w N y y w N x w N x N

2 6 6 1

where the stress resultants are referred to the fundamental state STABILITY ANALYSIS (continued)

MECHANICAL AND THERMAL BUCKLING

THE STRESS RESULTANTS IN THE PREBUCKLING STATE ARE ASSUMED TO BE CONSTANT.

THERMAL LOADS

cr cr cr

N N N N N N

3 6 2 2 1 1

, , α α α − = − = − = , ,

6 2 1

= = = N N N N N

T cr T cr

∫ −

+ + − − =

2 2

) , , ( ) )( ( ) 1 ( 1

h h m c cm m c cm T cr

dz z y x T f E f E N α α ν

• UNIFORM TEMPERATURE RISE:

cr i f

T T T z T = − = ) (

• LINEAR TEMPERATURE CHANGE ACROSS THE THICKNESS :

m cr

T h z T z T − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = 2 1 ) (

m c cr

T T T − =

WHERE:

MECHANICAL AND THERMAL BUCKLING

• NON-LINEAR TEMPERATURE CHANGE ACROSS THE THICKNESS:

SOLVING A SIMPLE STEADY STATE HEAT TRANSFER EQUATION

⎪ ⎩ ⎪ ⎨ ⎧ = = − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ −

c m

T h T T h T dz z dT z K dz d ) 2 ( , ) 2 ( , ) ( ) (

m c cm m c cm

K K K K f K z K − = + = ) (

∑

= +

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− + + =

5 ) 1 (

2 1 ) 1 ( 1 ) (

j jn j m cm cr m

h z K K jn D T T z T

SERIES SOLUTION (7 TERMS):

Solution of Energy Equation

∑

=

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛− + =

5

) 1 ( 1

j j m cm

K K jn D

m c cr

T T T − =

∑ ∑ ∑ ∑ ∑ ∑ ∑

= = = = = = =

= = = = = = =

m j j j m j j j m j j j m j j j m j j j m j j j m j j j

y x N y x N y x N y x N y x N w w y x N v v y x N u u

1 2 2 1 1 1 1 2 2 1 1 1 1 1 1

) , ( , ) , ( ) , ( , ) , ( , ) , ( , ) , ( , ) , ( ψ ψ ψ ψ ϕ ϕ ϕ ϕ INTERPOLATIONS

FINITE ELEMENT MODEL

1 3

ξ η

5 6 8 10 11 13 15 16 18 20 21 23 25 2 4 7 9 12 14 17 22 24 19

Q25 ( p=4)

1 5

ξ η

9 37 27 73 77 81 3 7 19 55 75 79 63 45

Q81 ( p=8)

41

ELEMENTS

, 1 , , 1 , ) ( ) ( ) ( , ) ( ) ( ) (

1 1 2 1 1 1

+ = − − = − − =

∏ ∏

+ ≠ = + ≠ =

p i L L

p i k k k i k i p i k k k i k i

K η η η η η ξ ξ ξ ξ ξ

LAGRANGE POLYNOMIALS

FINITE ELEMENT MODEL (continued)

SHAPE FUNCTIONS

. ) 1 )( 1 ( ), ( ) (

2 1

i p j k L L N

j i k

+ + − = = η ξ

[ ]{ } [ ]{ }

∆ = ∆

G

K K λ

EIGENVALUE PROBLEM

FINITE ELEMENT MODEL (continued)

NUMERICAL RESULTS

Comparison of the dimensionless critical load for symmetric three cross-ply laminated plates under uniaxial and biaxial compression, and shear loading (4×4Q25 full integration)

3 2 2

h E b N N

cr cr =

b a Theory Uniaxial Biaxial Shear LW3D [24] 22.2347 9.9424 8.8184 FSDT [25] 22.3151 10.2024

• Present TSDT

22.1164 9.9330 8.7369 1 Present FSDT 22.3151 10.2024 8.9672 LW3D [24] 16.4247 3.2694 3.1317 FSDT [25] 16.4340 3.2868

• Present TSDT

16.2986 3.2597 3.1285 2 Present FSDT 16.4340 3.2868 3.1579

h a S h E b N N

m cr cr

= = ,

3 2

Comparison of the uniaxial critical load for antisymmetric two cross-ply laminated square plates for various boundary conditions (2×2Q81 full integration)

3 2 2

h E b N N

cr cr =

a h Theory SSSS SCSC SFSF LW3D [24] 11.2560 19.5762 4.7662 TSDT [26] 11.562 21.464 4.940 FSDT [26] 11.353 20.067 4.851 Present TSDT 11.5193 21.0224 4.9185 0.1 Present FSDT 11.3526 20.0669 4.8507 LW3D [24] 8.0732 8.9584 3.4867 TSDT [26] 8.769 11.490 3.905 FSDT [26] 8.277 9.757 3.682 Present TSDT 8.6514 10.7516 3.8449 0.2 Present FSDT 8.2773 9.7566 3.6817

NUMERICAL RESULTS (continued)

Comparison of the critical temperature of FGM plates under nonlinear temperature rise (4×4Q25 full integration) ) 5 , 100 (

• m

cr

T h a T = = n Theory 1 = b a 2 = b a 3 = b a 4 = b a 5 = b a Lanhe [14] 24.1622 75.3952 160.5901 279.5281 431.8769 Javaheri [13] 24.1982 75.4955 160.9910 280.6848 434.5767 Present TSDT 24.1790 75.3752 160.5104 279.2983 431.3415 0.0 Present FSDT 24.1790 75.3752 160.5104 279.2981 431.3412 Lanhe [14] 7.6554 38.6328 90.1801 162.1757 254.4500 Javaheri [13] 7.6636 38.6838 90.3843 162.7649 255.8247 Present TSDT 7.6538 38.6226 90.1395 162.0586 254.1769 1.0 Present FSDT 7.6538 38.6226 90.1395 162.0585 254.1768 Lanhe [14] 4.8699 28.2918 67.2531 121.6415 191.3010 Javaheri [13] 4.8774 28.3389 67.4414 122.1849 192.5693 Present TSDT 4.8665 28.2705 67.1683 121.3975 190.7334 5.0 Present FSDT 4.8684 28.2824 67.2155 121.5334 191.0494

3 . ), 1 ( 10 23 , 204 , 70 3 . ), 1 ( 10 4 . 7 , 4 . 10 , 380

6 6

= × = = = = × = = =

− − m

• m

m m c

• c

c c

C mK W K GPa E C mK W K GPa E ν α ν α

ALUMINA-ALUMINIUM PLATES

) 5 , 10 (

• m

cr

T h a T = = n Theory 1 = b a 2 = b a 3 = b a 4 = b a 5 = b a Lanhe [14] 3256.310 7640.640 13853.53 20760.85 27586.74 Javaheri [13] 3409.821 8539.554 17089.10 29058.47 44447.67 Present TSDT 3227.363 7484.611 13337.61 19674.28 25731.38 0.0 Present FSDT 3227.248 7483.072 13327.95 19639.08 25641.06 Lanhe [14] 1976.297 4691.69 8619.42 13141.43 17740.25 Javaheri [13] 2055.00 5157.03 10327.07 17565.13 26871.21 Present TSDT 1961.329 4609.313 8348.69 12530.97 16663.80 1.0 Present FSDT 1961.269 4608.498 8343.41 12511.06 16611.01 Lanhe [14] 1481.30 3478.32 6288.94 9415.58 12481.59 Javaheri [13] 1553.34 3899.48 7809.73 13284.08 20322.53 Present TSDT 1450.99 3317.92 5789.28 8352.37 10707.20 5.0 Present FSDT 1467.68 3404.76 6053.10 8897.18 11587.11

NUMERICAL RESULTS (continued)

MECHANICAL BUCKLING

3 . ), 1 ( 10 10 , 09 . 2 , 151

6

= × = = =

− c

• c

c c

C mK W K GPa E ν α

Zirconia

Effect of the ratio S on the critical buckling load of FGM square plates under uniaxial compression Aluminium-zirconia Aluminium-alumina

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 10 20 30 40 50 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 10 20 30 40 50

Ceramic n = 0.2 n = 0.5 n = 1.0 n = 2.0 Metal

NUMERICAL RESULTS (continued)

n

cr

N

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 2 4 6 8 10 TSDT (S = 5) TSDT (S = 10) TSDT (S = 100) FSDT (S = 5) FSDT (S = 10) FSDT (S=100)

n

cr

N

4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 2 4 6 8 10 TSDT (S = 5) TSDT (S = 10) TSDT (S = 100) FSDT (S = 5) FSDT (S = 10) FSDT (S = 100)

Effect of the volume fraction exponent on the critical buckling load of FGM square plates under uniaxial compression

Aluminium-zirconia Aluminium-alumina

NUMERICAL RESULTS (continued)

Effect of the ratio S on the critical buckling temperature of FGM square plates under uniform temperature rise (aluminium-alumina)

) ( C T

• cr
• 1500

1500 3000 4500 6000 7500 9000 10500 12000 13500 15000 10 20 30 40 50 Ceramic n = 0.2 n = 0.5 n = 1.0 n = 2.0 Metal

S

Effect of the volume fraction exponent on the critical buckling temperature of FGM square plates under uniform temperature rise (aluminium-alumina)

) ( C T

• cr

n

1000 2000 3000 4000 5000 6000 2 4 6 8 10 TSDT (S = 5) TSDT (S = 10) TSDT (S = 20) FSDT (S = 5) FSDT (S = 10) FSDT (S=20)

NUMERICAL RESULTS (continued)

Effect of the ratio S on the critical buckling temperature of FGM square plates under nonlinear temperature change across the thickness (aluminium-alumina) Effect of the volume fraction exponent on the critical buckling temperature of FGM square plates under nonlinear temperature change across the thickness (aluminium-alumina)

) ( C T

• cr

S

• 1000

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 10 20 30 40 50 60 70 80 Ceramic n = 0.2 n = 0.5 n = 1.0 n = 2.0 Metal

) ( C T

• cr

n

2000 4000 6000 8000 10000 12000 2 4 6 8 10 TSDT (S = 5) TSDT (S = 10) TSDT (S = 20) FSDT (S = 5) FSDT (S = 10) FSDT (S=20)

NUMERICAL RESULTS (continued)

CONCLUSIONS

• A displacement finite element model for stability

problems is derived (shear locking is avoided).

• Comparisons of our results with other found in the

literature validate the present formulation.

• The effect of the shear deformation is significant,

especially for thick plates, hence it cannot be

• neglected. Finally, differences in the results of the

TSDT and FSDT are minor but more significant for FGM plates than those of homogeneous plates (i.e., fully ceramic or fully metal plates).

I thank …

• YOU for your interest in my

presentation

• Professors Glaucio Paulino,

Horacio Espinosa, Fernando Rochinha, and Ney Dumont for their kind invitation

• DULCE for her help with the flight

bookings, and for her excellent hospitality arrangements That which is not given is lost

CONCLUDING REMARKS CONCLUDING REMARKS

Numerical simulation continues to be a major component of

computer aided engineering and manufacturing.

Material modeling at different scales presents new challenges in

developing more sophisticated and accurate computational techniques.

A good understanding of (a) the physics and (b) the numerical

method being used to simulate the process is essential for an accurate and efficient solution. Thus, engineers with good background in particular engineering subjects as well as in computational mechanics will continue to have excellent opportunities.