Analysis of buckling behaviour of functionally graded plates

BUCKLING BEHAVIOUR OF FUNCTIONALLY GRADED PLATES

Guided by : Dr. Beena KP Assistant Professor

Byju V M2 Structural Engineering Roll No. 5

Functionally Graded Material (FGM)

• Two or multi layer composite plate

• Smooth and continuous variation of material composition and properties

• Avoids the disadvantages of composites

• Thermal stress relaxation

• Light weight

• High heat and wear resistance

• Breakage resistance

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Representation of modern material hierarchy 3

Functionally Graded Material (FGM)

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First developed in Japan in 1984

Thermal barrier for space plane project

History of FGM

Manufacturing of FGM

Shot peening

Ion implantation

Thermal spraying

Electrophoretic deposition

Chemical vapour deposition

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Classification and types of FGM

Classification

Natural FGM – Bone, bark, teeth

Bulk FGM

Wear resistant FGM

Others

Types

Ceramic - metal

Titanium alloy with graded density

Cemented carbide and titanium

Precious metals and metal oxides

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FGM

Nuclear reactor

Fusion pellets, Plasma wall

Space Applications

Rocket components, space plane frame

Medical Applications

Artificial bone, skin, dentistry

Communication

Optical fibre, lenses, semi conductors

Energy

Thermoelastic generators, solar cells, sensors

Others

Building materials, window glass, sports goods

Applications of FGM

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Applications of FGM

Rocket thrust chamber Artificial hip joint

High intensity discharge lamp

Metal - Ceramic FGM

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Ceramic

High thermal resistance

Low toughness and brittle

Cannot be directly used in engineering applications

Metals

Tough and Ductile

Ideal for engineering applications

FGM

The best of both

High strength

High temperature resistance

Analysis of FGM

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Classical Plate Theory

First, Second and third order shear deformation theory

Sinusoidal shear deformation theory.

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•Power law •Sigmoid function •Exponential function

Young’s modulus (E) – a function of composition and thickness

Mathematical idealization of material properties

PFGM SFGM EFGM

Variation of E

Power-law function Sigmoid function Exponential function

Volume fraction 𝑔 𝑧 =

𝑧 +ℎ2

ℎ

𝑝

𝑔1 𝑧 = 1 −1

2

ℎ2− 𝑧

ℎ2

𝑝

𝑔2 𝑧 = 1 −1

2

ℎ2+ 𝑧

ℎ2

𝑝

----------

Young’s modulus

𝐸 𝑧 = 𝑔 𝑧 𝐸1+ [1− 𝑔(𝑧)] 𝐸2

𝐸(𝑧) = 𝑔1(𝑧)𝐸1 + [1 – 𝑔1(𝑧)]𝐸2

𝐸(𝑧) = 𝑔2(𝑧)𝐸1 + [1 – 𝑔2(𝑧)]𝐸2

E(z) = A𝑒𝐵(𝑧+ℎ2)

where, A= E2

B=1

ℎ𝑙𝑛

𝐸1

𝐸2


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P-FGM

E-FGM

S-FGM


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• Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading (M. Mahdavian, 2009)

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• Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading (Samsam Shariat B.A. et. Al., 2005)

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Case studies

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Buckling of FGM rectangular plates under in-plane compressive loading

Fourier solution for the in-plane Airy stress field

Galerkin’s approach for stability analysis

Results for isotropic case validated with reference articles and FEM solution

Study of the effect of power law index in buckling of FGM plate

Case study 1 - Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading


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Rectangular plate under compressive loading in x – direction

solution of the two-dimensional elasticity problem governed by

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φ2 necessary to eliminate the residual shear stress on the

edges y = ±𝑏 2

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Approximate solution by Galerkin method

Trial function

Stability analysis


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Load cases considered

Total buckling coefficient for an isotropic square plate with simply supported edges

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No Load distribution Super

position

method

FEM

solution

1 Concentrated load 2.409 2.582

2 Triangular load 3.540 3.315

3 Uniform load 4.000 3.972

4 Reverse triangular load 4.690 4.810

5 Sinusoidal load 5.149 5.286


Total buckling coefficient for FGM square plate with simply supported edges for various p

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p Concentrated

load

Triangular

load

Uniform

load

Reverse

triangular

load

Sinusoidal

load

1 4.015 5.901 6.666 7.770 8.581

2 14.729 21.645 24.452 28.450 31.477

3 29.855 43.873 49.562 57.67 63.801

4 49.086 72.134 81.487 94.81 104.898

5 72.376 106.359 120.151 139.81 154.669

6 99.723 146.547 165.551 192.632 213.111

7 131.137 192.711 217.699 253.31 280.242

8 166.625 244.861 276.613 321.862 356.080

9 206.194 303.009 342.301 398.295 440.630

10 249.859 367.162 414.773 482.623 533.932

11 297.597 437.328 494.037 574.853 635.968

12 349.438 513.511 580.099 674.994 746.755

13 405.378 595.716 672.964 783.049 866.298

14 465.417 683.946 772.635 899.025 994.603

15 529.558 778.204 879.115 1022.92 1131.674


Buckling coefficient Kcr of FGM square plate, for simply supported edges, for

various load distributions and for various values of power law index. 21


Power law index (p)

22 Variation of Kcr of FGM rectangular plate with aspect ratio (a/b) for different loads

Concentrated load Sinusoidal Load

Triangular Load Reverse triangular Load


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Conclusions of study

1. The critical buckling coefficient (Kcr) for the FGM plate is generally higher than the isotropic rectangular plate.

2. The critical buckling coefficient for the FGM rectangular plate is increased by increasing the aspect ratio (a/b ).

3. The critical buckling coefficient for the FGM rectangular plate is increased by increasing the volume power law index p.

4. The analytical results will be useful for judging the accuracy of various approximate methods commonly employed.


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Rectangular imperfect plate made of functionally graded material with simply supported edge conditions and subjected to an in-plane loading in two directions

Classical plate theory

Power law function

Aluminium E = 7X104 N/mm2

Alumina E = 38X104 N/mm2

Simply supported on all edges

Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading

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The imperfections of the plate represented as,

μh − the amplitude of imperfection μ − between 0 and 1 m, n − number of half waves in x- and y-directions Approximate solution


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Em and Ecm - the Young’s moduli of the metallic and ceramic phases

p - the power law index.


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Critical buckling load


28 Pxc under uniaxial compression (R=0) Vs p and a/b

p=0

p

p=3

p=10


29 Pxc under uniaxial compression (R=0) Vs µ and a/b

p=1


30 Pxc under uniaxial compression (R=0) Vs p and b/h

p=0

p=1 p

p=10


31 Pxc under uniaxial compression (R=0) Vs p and µ

p=0

p=1

p=3

p=10


32 Pxc under biaxial compression (R=1) Vs p and a/b

p=0

p=1

p

p=10


33 Pxc under biaxial compression (R=1) Vs µ and a/b

p=1


34 Pxc under biaxial compression (R=1) Vs p and b/h

p=10

p

p=1

p=0


35 Pxc under biaxial compression (R=1) Vs p and µ

p=0

p=1

p=3

p=10


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Pxc under compression along x-axis and tension along

y-axis (R= -1)Vs p and a/b


p=0

p=1 p

p=10

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y-axis (R= -1)Vs µ and a/b


p=1

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y-axis (R= -1)Vs p and and b/h


p=0

p=1

p

p=10

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y-axis (R= -1)Vs p and µ


p=0

p=1

p=3

p=10

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Conclusions of case study 2

1.The buckling load of an imperfect functionally graded plate is greater than a perfect one.

2.The critical buckling load of a functionally graded plate increases with increasing imperfection amplitude µ.

3.The critical buckling load Pxc of an imperfect functionally graded plate is reduced when the power law index p increases.

4.The buckling mode of the plate may change with increasing the aspect ratio a/b.

5.The critical buckling load Pxc for the functionally graded plates generally decreases with the increase of aspect ratio a/b.


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Conclusions of case study 2

6. The critical buckling load Pxc for the functionally graded plates decreases with increasing dimension ratio b/h.

7. The critical buckling load Pxc for the plates under uni-axial compression are greater than the plates under biaxial compression.

8. The critical buckling load Pxc for the plates under combined compression and tension are greater than for plates under uniaxial and biaxial compression. This conclusion confirms that the addition of a tensile load in the transverse direction is seen to have a stabilizing influence.


Conclusions

1. Functionally graded materials having gradually varying material composition and mechanical properties across the thickness have superior properties of heat resistance and are suitable for use in extreme temperature conditions.

2. The variation of material property of FGM plate can be idealised by power law, sigmoidal or exponential functions, of which power law is the simplest.

3. Classical plate theory, used for the analysis of isotropic plates is applicable to FGM plates also.

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4. The critical buckling load of FGM plate can be estimated by approximate solution of the stability equation using appropriate Airy’s stress function by adopting a superposition approach.

5. The critical buckling coefficient of FGM plate increases with the power law index and also with the aspect ratio.

6. The critical buckling load of geometrically imperfect FGM plates depends on the load ratio and the amplitude of imperfection. In this case, the critical buckling load decreases with increase in power law index.

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Conclusions

References

1. Jha D.K., Tarun Kant, and Singh R.K. (2013), “A Critical Review of Recent Research on Functionally Graded Plates”, Composite Structures, Vol. 96, pp. 833–849.

2. M. Mahdavian (2009), Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading, Journal of Solid Mechanics Vol. 1, No. 3 pp. 213-225

3. Samsam Shariat B.A., Javaheri R. and Eslami M.R. (2005), “Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading”, Thin Walled Structures, Vol. 43, pp.1020-1036.

4. Shyang-Ho Chi and Yen-Ling Chung (2006), “Mechanical behaviour of functionally graded material plates under transverse load- Part I, Analysis”, International Journal of Solids and Structures, Vol. 43, No. 13, pp. 3657-3674.

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Technology

Analysis of buckling behaviour of functionally graded plates