analysis of buckling behaviour of functionally graded plates
DESCRIPTION
This deals with FGM plates in general with some case studies on their buckling behaviourTRANSCRIPT
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
Mathematical idealization of material properties
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P-FGM
E-FGM
S-FGM
Mathematical idealization of material properties
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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
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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Case study 1 - Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading
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
Case study 1 - Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading
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Case study 1 - Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading
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
Case study 1 - Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading
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
Case study 1 - Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading
Buckling coefficient Kcr of FGM square plate, for simply supported edges, for
various load distributions and for various values of power law index. 21
Case study 1 - Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading
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
Case study 1 - Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading
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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.
Case study 1 - Buckling Analysis of Simply-supported Functionally Graded Rectangular Plates under Non-uniform In-plane Compressive Loading
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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
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
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Em and Ecm - the Young’s moduli of the metallic and ceramic phases
p - the power law index.
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
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Critical buckling load
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
28 Pxc under uniaxial compression (R=0) Vs p and a/b
p=0
p
p=3
p=10
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
29 Pxc under uniaxial compression (R=0) Vs µ and a/b
p=1
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
30 Pxc under uniaxial compression (R=0) Vs p and b/h
p=0
p=1 p
p=10
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
31 Pxc under uniaxial compression (R=0) Vs p and µ
p=0
p=1
p=3
p=10
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
32 Pxc under biaxial compression (R=1) Vs p and a/b
p=0
p=1
p
p=10
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
33 Pxc under biaxial compression (R=1) Vs µ and a/b
p=1
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
34 Pxc under biaxial compression (R=1) Vs p and b/h
p=10
p
p=1
p=0
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
35 Pxc under biaxial compression (R=1) Vs p and µ
p=0
p=1
p=3
p=10
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
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Pxc under compression along x-axis and tension along
y-axis (R= -1)Vs p and a/b
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
p=0
p=1 p
p=10
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Pxc under compression along x-axis and tension along
y-axis (R= -1)Vs µ and a/b
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
p=1
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Pxc under compression along x-axis and tension along
y-axis (R= -1)Vs p and and b/h
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
p=0
p=1
p
p=10
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Pxc under compression along x-axis and tension along
y-axis (R= -1)Vs p and µ
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
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.
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
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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.
Case study 2 - Buckling of Imperfect Functionally Graded Plates under In-plane Compressive Loading
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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