Thermal Buckling and Elastic Vibration Analysis of Functionally Graded Beams and Plates Using Improved Third-Order Shear Deformation Theory

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Thermal buckling and elastic vibration analysis of functionally graded beams and plates using improved third-order shear deformation theory

A thesis

Nuttawit Wattanasakulpong

2012

Submitted in partial fulfillment of the requirements for

Doctor of Philosophy

School of Mechanical and Manufacturing Engineering,

The University of New South Wales

ORIGINALITY STATEMENT

I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project’s design and conception in style, presentation and linguistic expression is acknowledged.

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Date …16/07/2012……………………………………………………

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I certify that the Library deposit digital copy is a direct equivalent of the final officially approved version of my thesis. No emendation of content has occurred and if there are any minor variations in formatting, they are the result of the conversion to digital format.

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Abstract

Functionally graded materials (FGMs) have been developed for general purpose structural components such as rocket engine components or turbine blades where the components are exposed to extreme temperatures. The earliest FGMs were introduced by Japanese scientists in the mid-1980s as ultra-high temperature-resistant materials for aerospace applications. Recently, these materials have found other uses in electrical devices, energy transformation, biomedical engineering, optics, etc. FGMs are microscopically inhomogeneous spatial composite materials, typically composed of a ceramic-metal or ceramic-polymer pair of materials. Therefore, it is important to investigate the behaviors of engineering structures such as beams and plates made from FGMs when they are subjected to thermal and dynamic loads for appropriate design.

The material property profiles of FGMs vary across the graded direction. Therefore, using an improved third order shear deformation theory (TSDT) based on more rigorous kinetics of displacements to predict the behaviors of functionally graded beams and plates is expected to be more suitable than using other theories. Thus, in this research, the improved TSDT is used to investigate thermal buckling and elastic vibration response of functionally graded beams and plates.

For the first time in this research temperature dependent material property solutions, are adopted to investigate thermal buckling results of functionally graded beams and plates. Additionally, the research includes natural frequency and forced vibration analysis of functionally graded plates subjected to a uniformly distributed dynamic load acting over the plate domain. To obtain the solutions, the Ritz method using polynomial and trigonometric functions for defining admissible displacements and rotations is applied to solve the governing equations. The numerical results are validated by published and experimental results.

To clearly understand functionally graded materials beam specimens were manufactured from alumina-epoxy using a multi-step sequential infiltration technique. These beams were then subject to microscopic analysis to determine the profiles of the constituents. Finally v experiments were conducted to determine the vibration characteristics and the results were compared to analysis using the improved TSDT.

To compute theoretical parts in this research, the material compositions of the functionally graded beams and plates are assumed to vary smoothly and continuously throughout the thickness according to the power law distribution. Several significant aspects such as thickness and aspect ratios, materials, temperature, added mass etc. which affect analytical results are taken into account and discussed in detail.

The original work in this thesis includes the application of the improved TSDT to thermal buckling and elastic vibration problems of functionally graded beams and plates. New critical buckling temperature results for the case of temperature dependent material properties have been solved by an iterative calculation technique. The results reveal that the effect of temperature dependent material on reduced buckling temperatures is more profound for a thicker beam and plate than a thinner one. The relationship between the critical temperatures and natural frequencies of the beam and plate structures are also presented and discussed.

Acknowledgements

First of all, I would like to kindly acknowledge the financial support offered by the Mahanakorn University of Technology (MUT) scholarship. I wish to thank Prof. Variddhi Ungbhakorn who nominated and recommended me to receive the MUT scholarship.

My thanks go to many people who provided great support and had an important role in this research. I would like to express my gratitude to my supervisor, Asso. Prof. B. Gangadhara Prusty, and co-supervisors Prof. Donald W. Kelly and Prof. Mark Hoffman of the University of New South Wales (UNSW), for their continuous support and valuable guidance throughout this research.

I had also the opportunity to work with people in material science laboratory of UNSW. Therefore, my acknowledgments are extended to Dr. George Yang for his technical guidance and training. Dr. Auppatham Nakaruk is thanked for his comment and discussion on material fabrication. My thanks also go to Russell Overhall who helped and provided me a useful guidance of vibration test in mechanical laboratory. Thank you to everyone else who help me with this research.

Last but not least, I wish to profoundly thank my parents and my sisters for their unconditional love and unlimited support. Without their encouragement, I would not have been able to overcome many difficulties and challenges during this research.

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Table of Contents

Chapter 1 General introduction 1 1.1 Introduction of functionally graded materials 2 1.2 Problem definition 6 1.3 Objectives and scope of research 8 1.4 Thesis outline 8 1.5 List of publications 10 Chapter 2 Literature review 11 2.1 Functionally graded materials 12 2.1.1 Gradient description 13 2.1.2 Material properties of composite and functionally graded materials 17 2.2 Beam and plate theories 20 2.2.1 Beam theories 20 2.2.2 Layerwise theory for beam and plate analysis 24 2.2.3 Common plate theories and an improved TSDT 26 2.2.4 The refined plate theory 30 2.3 Bending analysis 32 2.3.1 Bending analysis of functionally graded beams 32 2.3.2 Bending analysis of functionally graded plates 35 2.4 Stability analysis 38 2.4.1 Stability analysis of functionally graded beams 38 2.4.2 Stability analysis of functionally graded plates 40 2.5 Vibration analysis 43 2.5.1 Vibration analysis of functionally graded beams 44 2.5.2 Vibration analysis of functionally graded plates 48 2.6 Functionally graded material fabrication 52 2.6.1 Thermal spraying technique 52 2.6.2 Powder metallurgy technique 56 2.6.3 Infiltration technique 58 2.7 Conclusion 62

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Chapter 3 Development of analytical method using improved TSDT 64 3.1 (a) Thermal buckling and elastic vibration analysis of FG beams 67 3.1.1 Strain energy for FG beams 68 3.1.2 The potential energy for FG beams due to thermal stresses 72 3.1.3 The kinetic energy for FG beams 74 3.1.4 The solution method for FG beam analysis 76 3.2 (b) Thermal buckling and elastic vibration analysis of FG plates 80 3.2.1 Strain energy for FG plates 82 3.1.2 The potential energy for FG plates due to thermal stresses 85 3.2.3 The kinetic energy for FG plates 88 3.2.4 Forced vibration analysis of FG plates 90 3.2.5 The solution method for FG plate analysis 90 Chapter 4 Thermal buckling and vibration analysis of FG beams and plates: Applications 95 4.1 Application of the improved TSDT to FG beam analysis 95 4.1.1 FG beam and material properties 96 4.1.2 Thermal buckling of FG beams based on the improved TSDT 99 4.1.3 Thermo-elastic vibration of FG beams based on the improved TSDT 106 4.2 Application of the improved TSDT to FG plate analysis 112 4.2.1 Thermal buckling of FG plates based on the improved TSDT 113

4.2.2 Thermo-elastic vibration of FG plates based on the improved TSDT 122 4.2.3 Thermo-elastic forced vibration of FG plates based on the improved TSDT 133 Chapter 5 FGM fabrication 140 5.1 Specimen Fabrication using multi-step sequential infiltration technique 140 5.1.1 Foam preparation 143 5.1.2 Slip casting and drying processes 145 5.1.3 Foam burning out and sintering 150 5.1.4 Polymer infiltration 153 5.2 Microstructure analysis 153 5.3 Alternative technique of specimen fabrication 157 Chapter 6 Results and discussion on the experimental and analytical predictions 163 6.1 Material properties of the layered FG beams 164 6.2 Vibration experiment of layered FG beams 166 6.3 The effect of added mass on natural frequencies on the layered FG beams 167 6.4 Free vibration analysis of continuous FG beams made of ceramic/polymer 175 Chapter 7 Conclusions 181 7.1 Thermal buckling and elastic vibration of FG beams 182 7.2 Thermal buckling and elastic vibration of FG plates 185 7.3 Further work 187 References 189

List of Figures

Fig. 1.1 The growing section of a tree stem (Emuna and Durban, 2010) 3 Fig. 1.2 An example of FGM application for aerospace engineering (Emuna and Durban, 2010) 4 Fig. 1.3 A C/C combustion chamber with a SiC/C FGM protective layer (Miyamoto et al., 1999) 5 Fig. 1.4 Schematic of the process for manufacturing diamond/SiC FGM cutting tools (Li, 1994, Miyamoto et al., 1999) 6 Fig. 2.1 Examples of FGMs: (a) a continuous FGM and (b) a layer-wise FGM 13 Fig. 2.2 Variations of the volume fraction of ceramics with various models based on the power law distribution 16 Fig. 2.3 Three-layered ceramic and metal FGM composite (Cho and Ha, 2002) 17 Fig. 2.4 Traditional composite material systems 18 Fig. 2.5 Cross-section displacements of three different beam theories (Wang et al., 2000, Yesilce and Catal, 2010) 23 Fig. 2.6 Schematic of thermal spraying technique for fabricating layered FGMs 53 Fig. 2.7 Geometry of multi-layered FGM beams (Kapuria et al., 2008a) 54 Fig. 2.8 Schematic of powder metallurgy technique to fabricate layered functionally graded beams made by Al/SiC 57 Fig. 2.9 Layered graded composite specimen made of alumina and epoxy: (a) photograph of the graded specimen: (b) a single interface between layers: (c) a series of layers with several distinct layers (Tilbrook et al., 2006) 60 Fig. 2.10 Processing sequences of infiltration technique using the GMFC 61 Fig. 3.1 The coordinate systems of beam and plate 65 Fig. 3.2 End conditions of FG beams 77 Fig. 4.1 Geometry of functionally graded beam 96 xi

Fig. 4.2 Variation of the volume fraction across thickness direction of the FG beams 97

ଶ ଶ Fig. 4.3 Dimensionless thermal buckling ሺߣൌȟܶ௖௥ܮ ߙ௠Ȁ݄ ሻ of C-C beams

made of Si3N4/SUS304 with n=0.3 103

ଶ ଶ Fig. 4.4 Dimensionless thermal buckling ሺߣൌȟܶ௖௥ܮ ߙ௠Ȁ݄ ሻ of

Al2O3/SUS304 beams with n=0.3 104

Fig. 4.5 Critical buckling temperature (K) of C-C beams made of Al2O3/SUS304 105 Fig. 4.6 Critical buckling temperature (K) of H-C beams with L/h=30 106

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.7 Dimensionless fundamental frequency ቆȳ ൌ ߱

of H-H beams made of Al2O3/SUS304 with L/h=30 derived from the solution I 108

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.8 Dimensionless fundamental frequency ቆȳ ൌ ߱

of C-C beams made of Al2O3/SUS304 with L/h=30 109

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.9 Dimensionless fundamental frequency ቆȳ ൌ ߱

of different boundary conditions for the FG beams made of Si3N4/SUS304 with L/h=30 and n=0.5 110

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.10 Dimensionless fundamental frequency ቆȳ ൌ ߱

of H-C beams made of Si3N4/SUS304 of the Solution II at ∆T=100 K 111

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.11 Dimensionless fundamental frequency ቆȳ ൌ ߱

of H-C beams made of various materials of the Solution II at ∆T=150 K with L/h=15 112 Fig. 4.12 FG plate geometry 113

Fig. 4.13 Critical buckling temperature (K) of Si3N4/SUS304 square plates with n=0.5: (a) SS-1 (b) CC 115 Fig. 4.14 Critical buckling temperature (K) of FG plates made out of xii

different pairs of materials (n=0.5, h/b=0.01) 118 Fig. 4.15 Critical buckling temperature (K) of FG plates made out of different pairs of materials (a/b=1.0, h/b=0.08) 119 Fig. 4.16 Critical buckling temperature (K) of fully clamped FG plates made of different pairs of materials; (a) effect of plate aspect ratio and (b) effect of the volume fraction index 122

ଶ ଶ ଵȀଶ Fig. 4.17 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯

of Si3N4/SUS304 plates (a/b=1.0; h/b=0.075; SS-2) 126

ଶ ଶ ଵȀଶ Fig. 4.18 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯

of Si3N4/SUS304 plates (a/b=1.0; h/b=0.10; SS-2; the Solution II) 127

ଶ ଶ ଵȀଶ Fig. 4.19 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯

of Si3N4/SUS304 plates 128

ଶ ଶ ଵȀଶ Fig. 4.20 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of FG plates (a/b=1.0; h/b=0.075; n=0.5) 129

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ܦFig. 4.21 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ

of Al2O3/SUS304 plates (n=0.5, a/b=1.0; a/h=10.0; SS-2; Solution II) 134

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ܦFig. 4.22 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ

of Al2O3/SUS304 plates (a/b=1.0; a/h=10.0; SS-2; ΔT=300 K; Solution II ) 135

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ܦFig. 4.23 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ of FG plates made of different pairs of materials (a/b=1.0; a/h=10.0; SS-2; ΔT=300 K; Solution II ) 136

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ܦFig. 4.24 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ

of Al2O3/SUS304 plates (n=0.5, a/b=1.0; a/h=10.0; CC; Solution II) 137

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ܦFig. 4.25 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ

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of Al2O3/SUS304 plates (a/b=1.0; a/h=10.0; CC; ΔT=300 K; Solution II ) 138 Fig. 5.1 Schematic of fabricating process of graded composite specimen 142 Fig. 5.2 Schematic of foam preparation for the porous network imprint 144 Fig. 5.3 The uniaxial hot press used to produce a series of different density foam pieces 145 Fig. 5.4 Chemical ingredients used to produce alumina slip for slip casting process and the amount of the ingredients measured by using the 4 digit weighing machine 147 Fig. 5.5 Schematic cross-section of slip casting experimental set up 148 Fig. 5.6 (a) The set of casting mould consisting of Perspex and Plaster of Paris block; (b) examples of sufficiently dry specimens 150 Fig. 5.7 Temperature profile for foam, binder and ash burn out 151 Fig. 5.8 Temperature profile for sintering of the alumina phase 152 Fig. 5.9 Furnace used for burning out foam and sintering ceramic phase 152 Fig. 5.10 Material compositions across the thickness direction of the specimen made of alumina and epoxy resin 154 Fig. 5.11 Image of material compositions of alumina and epoxy resin at the whole graded region 155

Fig. 5.12 Alumina volume fraction Vc )( of the specimen 156

Fig. 5.13 Microstructure of imperfect specimen with buffer layers and porosities 157 Fig. 5.14 (a) Mixing balls and a bottle containing alumina powder, ground mothballs and CMC bonder powder; (b) the ball milling machine used for mixing materials 158 Fig. 5.15 (a) Steel die; (b) press machine used to compact mixed powders 159 Fig. 5.16 Vacuum chamber used to speed up epoxy-resin infiltration 160 Fig. 5.17 Diagrams of producing layered graded specimen made of alumina and epoxy with the combined technique; (a) layer preparation process; xiv

(b) epoxy infiltration process 161 Fig. 5.18 Layered FGM specimen made of alumina and epoxy with porosities 162

Fig. 6.1 The volume fraction of alumina ሺܸ௖ሻ of the specimen and its predicted material profile (h=3 mm; n=1.3) 165 Fig. 6.2 Geometry of a layered functionally graded beam 166 Fig. 6.3 Material compositions across the layered FG beam thickness 166 Fig. 6.4 Experimental set up for vibration testing of the layered functionally graded beam 167 Fig. 6.5 Material compositions of FGM II (a) and FGM III (b) 170 Fig. 6.6 The fundamental frequency (kHz) of FGM III beams with the effect of added mass position (L/h=20, β=0.5, d=10 mm): (a) S-S (b) C-C (c) C-S and (d) C-F 174 Fig. 6.7 Volume fraction of ceramic across the beam thickness with different values of the volume fraction index 176 Fig. 6.8 Material properties of FG beams across the thickness with different values of the volume fraction index; (a) Young’s modulus (Pa), (b) density (Kg/m3) 177

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 6.9 Dimensionless fundamental frequencies ቆȳ ൌ ߱ ceramic/polymer FG beam with n=10.0 178 Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig 6.10 Dimensionless fundamental frequencies ቆȳ ൌ ߱

ceramic/polymer FG beams with various common boundary conditions 180

List of Tables

Table 2.1 General information of FGM types 62

Table 3.1 Displacement and rotation field indices for the Ritz method 78 Table 4.1 Temperature dependent coefficients of Young’s modulus

ܧ(Pa), thermal expansion coefficient ߙ (1/K), Poisson’s ratio ߥ, and mass density ߩ (kg/m3) for various materials (Shen, 2009) 99

ଶ ଶ Table 4.2 Dimensionless thermal buckling ሺߣൌȟܶ௖௥ܮ ߙଵȀ݄ ሻ of symmetric cross-ply beams (0o/90o/0o) with L/h=10 100

Table 4.3 Convergence studies for thermal buckling of Si3N4/SUS304 beams of temperature independent with n=0.5 and L/h=15 101

ଶ ଶ Table 4.4 Dimensionless thermal buckling ሺߣൌȟܶ௖௥ܮ ߙ௠Ȁ݄ ሻ of H-H beams with L/h=20 102

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮTable 4.5 Dimensionless fundamental frequency ቆȳ ൌ ߱

of Al/Al2O3 beams with n=0.3 under ambient temperature 107

Table 4.6 Comparisons of Critical buckling temperature (K) of Al2O3/Al plates based on the solution I 114

Table 4.7 Critical buckling temperature (K) of Al2O3/SUS304 plates (b/a=1.0) 116 Table 4.8 Critical buckling temperature (K) of FG plates made of different pairs of materials 117 Table 4.9 Critical buckling temperature (K) of fully clamped plates

made of Al2O3/SUS304 (b/a=1.0) 120 Table 4.10 Critical buckling temperature (K) of fully clamped FG plates made of different pairs of materials (h/b=0.025) 121

ଵȀଶ Table 4.11 Dimensionless frequencies ൫ȳ ൌ ݄߱ሺߩ௧Ȁܧ௧ሻ ൯ of Al2O3/Al plates under ambient temperature (a/b=1.0) 123

ଶ ଶ ଵȀଶ Table 4.12 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ xvi

of FG plates under ambient temperature (a=b=0.2, h=0.025) 124

ଶ ଶ ଵȀଶ Table 4.13 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯

of Si3N4/SUS304 plates (a/b=1.0; h/b=0.1) 125

ଶ ଶ ଵȀଶ Table 4.14 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯

of fully clamped FG plates made of Si3N4/SUS304 (a/b=1.0; h/a=0.1) 130

ଶ ଶ ଵȀଶ Table 4.15 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯

of fully clamped FG plates made of Si3N4/SUS304 (h/a=0.1; n=0.5) 131

ଶ ଶ ଵȀଶ Table 4.16 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯

of fully clamped FG plates made of Si3N4/SUS304 (a/b=1) 132

ଶ ଶ ଵȀଶ Table 4.17 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of fully clamped FG plates made of different pairs of materials (a/b=1; h/a=0.1) 132

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ܦTable 4.18 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ of isotropic and FG plates (a/b=1.0; a/h=10.0; SS-2) 133

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ܦTable 4.19 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ

of Si3N4/SUS304 plates (a/b=1.0; a/h=10.0; ΔT=300 K; SS-2) 134

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ܦTable 4.20 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ of fully clamped FG plates made of different pairs of materials (a/b=1.0; a/h=10.0; ΔT=300 K; Solution II) 139 Table 6.1 Comparisons between theoretical and experimental fundamental

frequency ሺ݂ൌ߱Τሻ ʹߨ of the layered FG beam 169 Table 6.2 Details of three different types of the layered FG beams 170 Table 6.3 The fundamental frequency (kHz) of different types of FG beams without added mass effect 171 Table 6.4 Frequency results (kHz) of the layered FG beams

with added mass effect 171

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Table 6.5 Frequency results (kHz) of the layered FG beams with various mass ratios (L/h=20) 172

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮTable 6.6 Dimensionless frequencies ቆȳ ൌ ߱

of ceramic/polymer FG beams with L/h = 20 based on TSDT 179

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮTable 6.7 Dimensionless fundamental frequencies ቆȳ ൌ ߱

of ceramic/polymer beams with n = 2.0 based on TSDT 179

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List of Symbols

ߪ௜௝ Normal stress components

் ߪ௜௝ Thermal stress components

߳௜௝ Normal strain components

߬௜௝ Shear stress components

ߛ௜௝ Shear strain components

ܳ௜௝ The plane stress reduced elastic constants

ܧ Young’s modulus

ߩ Material density

ߙ The coefficient of thermal expansion

ߥ Poisson’s ratio

݊ The power law or the volume fraction index

ܸ௖ The volume fraction of ceramic

ܸ௠ The volume fraction of metal

ܶ଴ Ambient temperature at 300 degree Kelvin (K)

ȟܶ Temperature change

ݑǡ ݒ In-plane displacements of x and y axes

ݓ Transverse displacement of z axis

߶௫ǡ߶௬ Rotation functions of x and y axes

ܰ௫௫ǡܰ௬௬ǡܰ௫௬ Force resultants

ܯ௫௫ǡܯ௬௬ǡܯ௫௬ Moment resultants

ܲ௫௫ǡܲ௬௬ǡܲ௫௬ Higher-order stress resultants

ܳ௫ǡܳ௬ Shear force resultants

ܴ௫ǡܴ௬ Higher-order shear force resultants xix

ܧ௧ǡߙ௧ǡߩ௧ Young’s modulus, the coefficient of thermal expansion and material density at the top surface respectively

ܧ௕ǡߙ௕ǡߩ௕ Young’s modulus, the coefficient of thermal expansion and material density at the bottom surface respectively

ܣ௜௜ Extensional stiffness components

ܤ௜௜ Extensional-bending coupling stiffness components

ܦ௜௜ Bending stiffness components

ܧ௜௜ Extensional-warping coupling stiffness components

ܨ௜௜ Bending-warping coupling stiffness components

ܪ௜௜ Warping-higher order bending coupling stiffness components

ܣ௜௝ǡܤ௜௝ǡܦ௜௝ǡܧ௜௝ǡܨ௜௝ǡܪ௜௝ Shear stiffness components

் ் ் ் ் ் ܣ௜௜ǡܤ௜௜ǡܦ௜௜ǡܧ௜௜ǡܨ௜௜ ǡܪ௜௜ Material stiffness components due to thermal stresses

ܫ଴ǡܫଵǡܫଶǡܫଷǡܫସǡܫ଺ Moment of inertia components

ܷ௘ The total strain energy

ܸ௘ The potential energy due to thermal stresses

ܶ௘ The kinetic energy

ܶ௔ The kinetic energy due to added mass

ܹ௘ Work done

ȫ The total energy functional

ߚ The mass ratio

ܲ଴ The uniformly distributed force

ȟܶ௖௥ The critical buckling temperature (K)

߱ Natural frequency (rad/s)

߱௘௫ Frequency of external excitation

ߜ A given error tolerance

ߣ Dimensionless thermal buckling

ȳ Dimensionless frequency

ݓഥ Dimensionless deflection

xxi

Chapter 1 General introduction

______

Advances in material manufacturing technology have been implemented to develop new generation materials in order to fulfil a requirement of use. Since the history of human civilization evolved from stone tools to the Steel Age and to Space Age, an innovation of certain materials in each age was created to support technology development. Rails and steel boilers of the Industrial Revolution were brought in during the Steel Age. In the Space Age, composite materials were introduced for making smart structures that were strong and lightweight. Composite materials are known as the modern materials which are composed of two or more different materials, to have the desired properties in specified applications. The lightweight composite materials known as fiber-matrix laminated composites have been used successfully in aircraft, automotive, marine industries and other engineering applications. However, the mismatch in mechanical properties across the interface of two different materials may cause large inter-laminar stresses. Consequently, de-bonding and delamination problems can occur, especially in a high temperature environment. In general, various kinds of isotropic homogeneous materials such as the materials in the group of metals and polymers have been used widely in several engineering fields for many decades. Materials in the group of metals excel in high strength and toughness, while polymers are good in high flexibility and corrosive resistance. However, these kinds of materials fail to withstand extreme temperature loads. Therefore, to improve properties in terms of thermal resistance, materials in the group of ceramics can be used to mix with metals and polymers in order to combine their specific advantages. Owing to the recent growth of a tendency to use materials for marking engineering structures which are subjected to mechanical loads

Chapter 1 under high temperature environment, it is important to create a new class of materials in order to support such a requirement.

1.1 Introduction of functionally graded materials

Functionally graded materials (FGMs) are advanced materials which have spatially varying properties. Due to FGMs being involved in the classification of composite materials, the material compositions of FGMs are assumed to vary smoothly and continuously throughout the gradient directions. The earliest FGMs were introduced by Japanese scientists in the mid-1980s as ultra-high temperature resistant materials for aerospace applications (Koizumi, 1993, Niino et al., 1987). Recently, these materials have found other uses in electrical devices, energy transformation, biomedical engineering, optics, etc. (Suresh and Mortensen, 1998). Suresh and Mortensen (Suresh and Mortensen, 1998) also provided some introduction about the fundamentals of FGMs. At the introduction of FGMs, most of the essential concepts and information about the materials were largely unknown outside of Japan. The first book of FGMs written in English was published in London, U.K. (Miyamoto et al., 1999), and contained comprehensive explanations of fundamentals, manufacturing processes, design and the current applications of FGMs, which were useful and available for general researchers outside of Japan.

The growing mechanism of plants was a source of inspiration for FGMs. The transverse section of a tree stem is shown in Fig. 1.1 which presents the growth direction from the inside to the outside surface. In general, the growing mechanism of a tree is governed by the ability of the cells to detect stresses. Thus the outside surface is much stronger in order to protect the softer part inside the tree which is used for water absorption.

Chapter 1

Fig. 1.1 The growing section of a tree stem (Emuna and Durban, 2010)

The main purpose of FGM development is to produce extreme temperature resistant materials so that ceramics are used as refractories (materials with excellent resistance to heat) to mix with other materials, in order to create FGMs. However, ceramics themselves cannot be used to make engineering structures subjected to high amounts of mechanical loads. This is due to poor property of ceramics in toughness, with the result that other materials having a good toughness property, e.g. metals and polymers, are needed to mix with ceramics in order to combine the advantages of each material.

An example of FGMs used for a re-entry vehicle is shown in Fig. 1.2. The FGMs can be used to produce the shuttle structures. The heat source is created by the air friction of high velocity movement. If the structures of the vehicle are made from FGMs, the hot air flow is blocked by the outside surface of ceramics and transfers slightly into the lower surface. Consequently, the temperature at the lower surface is much reduced, which therefore prevents or minimises structural damage due to thermal stresses and thermal shock.

Chapter 1

Fig. 1.2 An example of FGM application for aerospace engineering (Emuna and Durban, 2010)

For other applications, FGMs can be used for a variety of potential applications in transport systems, energy conversion systems, cutting tools, machine parts, semiconductors, optics, biosystems, etc. Different potential applications require different key issues. For example, in aerospace and nuclear energy applications, the key issue is reliability rather than cost. Hence, FGMs used in these applications could be produced from a high quality of material constituents in order to have the combinations of incompatible functions such as refractoriness with toughness or chemical inertness with toughness. On the other hand, for applications of cutting and engine components, the main issue is to use FGMs to satisfy the cost/performance ratio reliability. The requirements of FGMs for these applications are wear, heat, and corrosive resistances as well as high strength of the materials.

Chapter 1

A carbon/carbon (C/C) composite combustion chamber with a SiC/C FGM protective layer was developed for the engine of the reaction control system of the Japanese space shuttle. The chamber was tested to examine the performance of heat resistance by using a hot gas flow test. The schematic of the combustion chamber is shown in Fig. 1.3.

Fig. 1.3 A C/C combustion chamber with a SiC/C FGM protective layer (Miyamoto et al., 1999)

Diamond cutting tools can be used for machining soft components, e.g. plastic contact lenses, polygonal mirrors of laser printers, hard disk substrates, with high precision. However, using conventional diamond cutting tools which are manufactured by joining a diamond crystal onto a metallic alloy shank with a silver solder containing active metals may lead to poor accuracy in machining. This is due to the silver solder’s lack of stiffness, which causes vibrations during machining. Thus the extremely stiff FGM diamond tools with a graded layer of diamond/SiC between the diamond chip and the SiC shank were developed to solve such problems in 1992 (Li, 1994, Miyamoto et al., 1999).

Chapter 1

Fig. 1.4 Schematic of the process for manufacturing diamond/SiC FGM cutting tools (Li, 1994, Miyamoto et al., 1999)

1.2 Problem definition

It is important to investigate and understand behaviour of the functionally graded materials subjected to various mechanical loads, for appropriate design. Beams and plates are often found in general engineering structures. Therefore, a beam or plate shaped FGM is an important structural element so as to be specifically focused in the current research. Functionally graded beams and plates subjected to static and dynamic loads under thermal environment need to be well-designed in order to have strong structures and to minimise cost required for their processing and manufacturing. To investigate such structures made 6

Chapter 1 of FGMs using the classical beam and plate theories, it is generally found that the deflection results of bending analysis are under-estimated, whereas, critical loads and natural frequencies of stability and dynamic analysis are over-estimated. Thus theories that take into account shear deformation effects are recommended to apply for analysing beams and plates produced from FGMs, in order to achieve more accuracy in predictions.

The first order shear deformation theory (FSDT) is used when the normal to the plate mid- plane before deformation remains straight but not necessarily normal to the deformed mid- plane. The higher order shear deformation theories (HSDTs) are needed when normal to the plate mid-plane before deformation does not remain straight after deformation; causing a nonlinear variation of displacements u and v over the beam and plate thickness. In the case of FGMs, due to the approximated properties across the thickness, the improved TSDT provides a better representation of the cubic variation of displacements through the thickness over the FSDT. The study presented by Shi (Shi, 2007) discussed the importance of using improved TSDT over other HSDTs for the accurate prediction of beams and plates. The analytical method described in this research uses the governing equation derived from the improved TSDT and solved by the Ritz method. Appropriate admissible functions used for defining displacements and rotations are employed in the Ritz method to obtain accurate solutions with minimal computational effort.

In order to investigate thermal buckling and vibration analysis of FGM structures under a high temperature environment, to satisfy a more realistic situation, the effective material properties of FGM could be a function of temperature. This can be explained that the materials become softer when they are in a higher temperature environment due to their elasticity modulus being reduced and their thermal expansion coefficient increased (Shen, 2009). As a result, temperature dependent material properties are implemented to compute the analytical results in this research.

Chapter 1

1.3 Objectives and scope of research

The objective of this research is to investigate thermal buckling and elastic vibration response of functionally graded beams and plates. An improved third order shear deformation theory is employed to analyse and describe behaviour of the beams and plates subjected to thermal loads. Two types of solutions, which are temperature independent material property and temperature dependent material property solutions, are adopted to find out thermal buckling results and natural frequencies of functionally graded beams and plates under a thermal environment. Additionally, forced vibration analysis of functionally graded plates subjected to a uniformly distributed dynamic load over the plate domain is also included in this research. The Ritz method using polynomial and trigonometric functions for defining admissible displacements and rotations is applied to solve the governing equations of the functionally graded beams and plates. The numerical results are validated by some available results and experimental results. To clearly understand the process sufficient for the prediction of natural frequencies of functionally graded beams, the beam samples made of alumina-epoxy are fabricated in order to test their vibration characteristics. To compute theoretical parts in this research, the material compositions of the functionally graded beams and plates are assumed to vary smoothly and continuously throughout the thickness according to the power law distribution.

1.4 Thesis outline

This research is to develop theoretical formulations based on the improved third order shear deformation theory to investigate thermal buckling and elastic vibration of functionally graded beams and plates. Several aspects, which have significant influences on the analytical results, are taken into investigation with some numerical and experimental validations. This thesis contains 7 chapters to describe the whole procedure of development and investigation, which have details as follows:

¾ Chapter 1 introduces general information about functionally graded materials and potential applications as well as a brief discussion of thesis objectives.

Chapter 1

¾ In Chapter 2, more details of the materials, microstructure analysis including the effective techniques used to produce the materials in the past are reviewed and presented. In terms of mechanics of the materials, the literature review focuses mostly on the topics that are useful and relevant to this research such as bending, buckling and vibration analysis of laminated composite and functionally graded beams and plates.

¾ Based on the improved third order shear deformation theory, the development of analytical method using the theory is described in Chapter 3, for thermal buckling and free and forced vibration analyses of functionally graded beams and plates. The Ritz method is adopted to solve the governing equations of beams and plates derived from energy approach. An iterative procedure used to obtain the temperature dependent material property solution is also presented for thermal buckling analysis.

¾ Chapter 4 provides the analytical results of thermal buckling, free and forced vibration response of functionally graded beams and plates with various boundary conditions under high temperature environment, which are derived from the theoretical formulations as presented in Chapter 3. Several aspects such as boundary conditions, thickness ratio, aspect ratio, temperature, materials, etc. which have considerable impact on the analytical results are taken into investigation. The significant discussion on the behaviour of the functionally graded beams and plates based on the investigated results is included in this chapter as well.

¾ The procedures used for producing functionally graded specimens, using a multi- step sequential infiltration technique, are explained in detail, which can be seen in Chapter 5. Within this chapter, the microstructure of the specimens is also investigated and discussed. An alternative technique of specimen fabrication developed by using the principles of other available techniques is also introduced and used to produce the specimens.

Chapter 1

¾ Chapter 6 presents results and discussion on the experimental and analytical predictions of vibration characteristics of functionally graded beams. The beam samples produced from alumina and epoxy, by using the fabricating technique as presented in Chapter 5, are used for vibration testing with various boundary conditions. The experimental results are compared with the theoretical predictions.

¾ Finally, in Chapter 7, a summary of the investigation and the important conclusions of this research are presented. The further work related to this research is suggested for future development and investigation. The details of the research papers based on the present research work reported in the thesis are presented below.

1.5 List of publications

1. (2010) Wattanasakulpong, N., Prusty, B. G., Kelly, D. W. and Hoffman, M.; A Theoretical investigation on the free vibration of functionally graded beams, CST 2010 Computational Structures and Technology, Sept 14-17, 2010, Valencia, Spain.

2. (2011) Wattanasakulpong, N., Prusty, B. G., and Kelly, D. W.; Thermal buckling and elastic vibration of third-order shear deformable functionally graded beams, International Journal of Mechanical Sciences. 53(9), 734-743.

3. (2012) Wattanasakulpong, N., Prusty, B. G., Kelly, D. W. and Hoffman, M.; Free vibration analysis of layered functionally graded beams with experimental validation, Journal of Materials and Design, 36:182-190.

4. Wattanasakulpong, N., Prusty, B. G., and Kelly, D. W.; Temperature dependent buckling and free vibration of functionally graded plates using an improved third order shear deformation theory, submitted to European Journal of Mechanics-A/Solids.

5. (under preparation) Wattanasakulpong, N., Prusty, B. G., and Kelly, D. W.; Free and forced vibration analysis of functionally graded plates under high temperature environment.

Chapter 2 Literature review

This chapter presents the literature review of traditional and new classes of composite materials used to make engineering structures. It is important to understand the behaviour of composite structures which are subjected to static and dynamic loads, for structural designs. The main focus of this chapter is to review previous investigations on a class of composite material, namely, functionally graded material. However, some useful researches dealing with traditional composite materials are also included, which can be used as a background for new composite material development.

To predict the behaviour of composite structures under various kinds of mechanical loads, several theories have been developed to improve accuracy in predictions. Some theories were chosen to investigate bending, buckling and vibration analysis of composite structures by many researchers in the past, including theoretical and experimental validations. For functionally graded materials, these materials were developed for making engineering components which are generally subjected to mechanical loads under high temperature environment. Therefore, thermal effects on material properties and structural behaviour have been taken into consideration in some previous investigations. An overview of existing techniques used for manufacturing functionally graded materials is also reviewed and presented in this chapter. The review contents which are relevant and useful for further investigation of this research can be expressed as follows.

Chapter 2

2.1 Functionally graded materials

2.2 Beam and plate theories

2.3 Bending analysis

2.4 Stability analysis

2.5 Vibration analysis

2.6 Functionally graded material fabrication

2.7 Conclusion

2.1 Functionally graded materials

Functionally graded materials (FGMs) are the new generation of composite materials which are usually produced from two different materials. FGMs can be considered as heterogeneous composites that are an optimal mixture of two material phases. The patterns of material mixture in FGMs are different compared to those of traditional composites like fiber-matrix laminated composites that consist of a filamentary phase embedded in a matrix phase. Based on functional performance requirements from point to point within the body of FGMs, the volume fraction and micro-structural morphology of the material compositions can be assumed to change continuously and smoothly along the graded direction. According to the continuous changes in the material compositions, these lead to the continuous variation of material properties from one surface to another. The gradation in the material properties can reduce thermal stresses, residual stresses and stress concentration factors. Consequently, this can eliminate the inter-laminar stresses between discrete materials often found in the laminated composites that may lead to de-bonding, delamination, plastic deformation and cracking problems, especially in high temperature applications.

FGMs are able not only to enhance strength but to improve the capabilities in terms of thermal barrier, wear and corrosive resistances. The earliest FGMs were introduced by Japanese scientists in the mid-1980s as ultra-high temperature-resistant materials for 12

Chapter 2

aerospace applications. Recently, these materials have found other uses in electrical devices, energy transformation, biomedical engineering, optics, etc. (Suresh and Mortensen, 1998).

2.1.1 Gradient description

There are many approaches used to describe the material gradient of FGMs which are manufactured from two phases of materials. In general, most of the approaches are based on the volume fraction distribution rather than developed from actual graded microstructures (Bao and Wang, 1995, Shen, 2009). Two types of FGMs, which account for continuous variation of material compositions across the graded direction shown in Fig. 2.1 (a) and for layer-wise variation of material compositions shown in Fig. 2.1 (b), were found in the literature survey. By considering the FGMs made of two distinct materials as shown in Fig. 2.1, it is assumed that the material compositions of the FGMs are varied from the top surface of Material A-rich to the bottom surface of Material B-rich. It can be seen that, for a case of continuous FGMs, the material compositions are changed with smooth distribution across the graded direction. However, for another case that can be defined as quasi-homogeneous material layer, the compositions are presented in the form of layered distribution.

Material A Material A

Material B Material B (a) (b)

Fig. 2.1 Examples of FGMs: (a) a continuous FGM and (b) a layer-wise FGM

Chapter 2

The values of material properties of the continuous FGMs can be predicted by using various mathematical models. As stated previously, FGMs are microscopically inhomogeneous spatial composite materials, hence, the material properties of FGMs depend on position. An exponential function used for defining material properties was applied in many studies in order to investigate crack propagation, fracture mechanics, bending, buckling and vibration (Erdogan, 1995, Sankar, 2001, Yang and Chen, 2008). The shear modulus G, Young’s modulus E and mass density ρ vary along the thickness direction (z) can be defined as the exponential distribution below:

ఉ௭ ఉ௭ ఉ௭ (଴݁ ǡߩሺݖሻ ൌߩ଴݁ (2.1ܧሺݖሻ ൌܧ଴݁ ǡܩሺݖሻ ൌܩ where ܩ଴ , ܧ଴ and ߩ଴ are the values of the shear modulus, Young’s modulus and mass density at the mid-plane (z=0), respectively. It is noted that β is a constant parameter to define the material property variation along the thickness direction, and FGMs can specialize in isotropic homogeneous materials by setting β=0.

On the basic of the rule of mixture, the effective material properties, P, can be written as

ܲൌܲ௠ܸ௠ ൅ܲ௖ܸ௖ (2.2) where Pm, Pc, Vm and Vc are the material properties and the volume fraction of the metal and ceramic, respectively, the compositions represent in relation to

ܸ௠ ൅ܸ௖ ൌͳǤ (2.3)

Another form of mathematical model, called a power law distribution has been used widely in a number of research investigations, especially for the mechanical engineering field (Cheng and Batra, 2000, Yang and Shen, 2003a, Navazi et al., 2006, Ganapathi, 2007, Sun and Luo, 2011). The power law distribution based on the rule of mixture was introduced by Wakashima et al. (Wakashima et al., 1990) in order to define the effective material properties of FGMs. The volume fraction of ceramic (Vc) can then be written as follows:

(ݖ ͳ ௡ (2.4 ܸ ൌ൬ ൅ ൰ ௖ ݄ ʹ

Chapter 2

where the positive number n ሺͲ൑݊൒λሻ is the power law or the volume fraction index. z is a distance parameter along the graded direction, while, h is the total length of the direction. To find out the results of material properties according to the power law distribution, this can be achieved by substituting the equations of material volume fractions in Eqs. (2.3 and 2.4) into Eq. (2.2).

In the study of Sofiyev (Sofiyev, 2009), several types of the power law distribution introduced by Pitakthapanaphong and Busso (Pitakthapanaphong and Busso, 2002) were used to define material properties of FGMs for investigating vibration and stability behaviour of FGM conical shells subjected to external pressure. The equations of material volume fraction used to calculate for ceramic (Vc) corresponding to different types of the power law distribution are expressed as:

(Linear type: ܸ௖ ൌݖ൅ͲǤͷ (2.5a ଶ (Quadratic type: ܸ௖ ൌ ሺݖ൅ͲǤͷሻ (2.5b ଶ (Inverse quadratic type: ܸ௖ ൌͳെሺͲǤͷ െ ݖሻ (2.5c ଶ ଷ (Cubic type: ܸ௖ ൌ͵ሺݖ൅ͲǤͷሻ െʹሺݖ൅ͲǤͷሻ . (2.5d

To understand clearly about the mathematical models presented in Eq. (2.5), the volume fractions of ceramic (Vc) in relation to the models plotted against the thickness ratio (z/h) are illustrated in Fig. 2.2.

Chapter 2

Volume fraction of ceramics

Dimensionless thickness, z/h Fig. 2.2 Variations of the volume fraction of ceramics with various models based on the power law distribution

Cho and Ha (Cho and Ha, 2002) introduced the mathematic model for predicting the material volume fraction of FGMs composed of three distinct layers. The top and bottom layers were respectively made by homogeneous ceramic and metal whereas the middle layer was given for the graded region as shown in Fig. 2.3. In this figure, the material volume fractions were varied in the y-axis which can be expressed as:

(ͳ െ ݀ ൑ ݕ ൑ െ݀௅ (2.6 ௠ ǡെ ݀௅ ൑ݕ൑െ݀௎ܥ௠ሺݕሻ ൌ൝ܸ Ͳ െ ݀௎ ൑ݕ൑െ݀

(௖ሺݕሻ ൌͳെܸ௠ሺݕሻ (2.7ܸ

௠ሺݕሻ is a continuous function that can satisfy the continuity conditions at lower andܥ where upper layer interfaces.

Chapter 2

Fig. 2.3 Three-layered ceramic and metal FGM composite (Cho and Ha, 2002)

2.1.2 Material properties of composite and functionally graded materials

A structural composite is a material system consisting of two or more constituents. Composite materials have been produced in order to obtain optimal material properties for specific application. Unlike isotropic-homogeneous materials, the mechanical performance and material properties of composite materials are dependent on several aspects, e.g. the properties of the constituents, geometry and the distribution of the phases. Therefore, the composite materials can be classified in the group of in-homogeneous and anisotropic materials. Several important features such as volume, weight, functional properties, strength, durability and cost need to be considered when producing composite materials. Various types of traditional composite materials found in the literature have been created for use in different purposes and applications. Examples of the composite materials, which are particle-matrix, discontinuous fibers or whiskers-matrix and continuous fibers-matrix, are shown in Fig. 2.4.

Chapter 2

Discontinuous fibers or Continuous Particle-Matrix whiskers-Matrix fibers-Matrix

Particle composite Unidirectional discontinuous Unidirectional continuous fiber composite fiber composite

Randomly oriented Cross-ply or fabric discontinuous continuous fiber composite composite

Quasi-isotropic composite

Fig. 2.4 Traditional composite material systems

Using the power law distribution and the rule of mixture, one can obtain the equations used for calculating material properties across the graded direction. It can be seen that all of the material property equations are a function of the volume fraction index (n) which is used to indicate percentages of material compositions. (Gibson et al., 1995).

(ݖ ͳ ௡ (2.8a ǡ ܧሻ ൬ ൅ ൰ ൅ ܧെ ܧሺݖሻ ൌ ሺܧ ௧ ௕ ݄ ʹ ௕ (ݖ ͳ ௡ (2.8b ߥሺݖሻ ൌ ሺߥ െߥ ሻ ൬ ൅ ൰ ൅ߥ ǡ ௧ ௕ ݄ ʹ ௕ (ݖ ͳ ௡ (2.8c ߙሺݖሻ ൌ ሺߙ െߙ ሻ ൬ ൅ ൰ ൅ߙ ǡ ௧ ௕ ݄ ʹ ௕ (ݖ ͳ ௡ (2.8d ߢሺݖሻ ൌ ሺߢ െߢ ሻ ൬ ൅ ൰ ൅ߢ ǡ ௧ ௕ ݄ ʹ ௕

Chapter 2

(ݖ ͳ ௡ (2.8e ߩሺݖሻ ൌ ሺߩ െߩ ሻ ൬ ൅ ൰ ൅ߩ Ǥ ௧ ௕ ݄ ʹ ௕

It is denoted that the typical values of material properties in Eq. (2.8) are Young’s modulus ሺݖሻ (in GPa), Poisson’s ratio ߥሺݖሻ, thermal expansion coefficient ߙሺݖሻ (in K-1), theܧ .thermal conductivity ߢሺݖሻ (in W mK-1) and material density ߩሺݖሻ (in Kg/m3), respectively The subscript t and b represent the material on the top and bottom surfaces.

However, it seems to be difficult to find out precise information about the size, the shape and the distribution of particle phases in FGMs. Different FGM fabrication techniques may give different graded patterns. As a result, the effective material properties at the graded region must be evaluated using mathematical models based on the volume fraction distribution and the approximate shape of the dispersed phases. The Mori-Tanaka scheme (Mori and Tanaka, 1973, Benveniste, 1987, Shen, 2009) was one of useful models to evaluate the effective moduli of FGMs. This scheme was more suitable when using with a typical FGM which has continuous matrix and discontinuous particle phases. The Mori- Tanaka scheme was constructed by taking into account the interaction of neighboring phases in terms of the elastic fields and the relationship of the volume fraction of the particulate phases. Many researchers have concentrated their attention on applying this scheme for investigating mechanical behaviour of the FGMs in various mechanical problems such as bending and vibration analysis (Wu and Chen, 2006, Ansari et al., 2011, Rezaei Mojdehi et al., 2011). In the Mori-Tanaka scheme, the matrix phase is denoted by subscript 1, and subscript 2 is given for the reinforced phase of spherical particles. The relationship of the volume fraction between two phases can be explained as ܸଵ ൅ܸଶ ൌͳ. Some, the material properties using the Mori-Tanaka scheme are given in (Shen, 2009).

In addition, another mathematical method that was used successfully to predict effective material properties of composite materials, consisted of each reinforcement inclusion embedded in a continuum material. The method is called “The self-consistent method” and was initially introduced by Hill (Hill, 1965). The basic idea of the method can be explained in that the phases of matrix and reinforcement are interchanged without any space between the material phases, and the material compositions are required to obey the rule of mixture. 19

Chapter 2

As a result, for some special types of composite materials, their material properties can be predicted effectively using the self-consistent method. In order to determine the effective moduli of composites which are constructed by interconnected skeletal microstructures, the self-consistent method could be recommended to deal with such problems.

According to the studies of Reuter et al. (Reuter et al., 1997) and Reuter and Dvorak (Reuter and Dvorak, 1998), the comparison between the Mori-Tanaka and self-consistent models for FGMs was investigated and demonstrated using the finite element simulation. It was found that a well-defined continuous matrix and discontinuous inclusions were predicted accurately using the Mori-Tanaka model. However, the self-consistent method was more appropriate when required for skeletal microstructures characterized by a wide transition zone between the regions with predominance of one of the constituent phases.

To avoid the effects of extensional-bending, extensional-warping, warping-bending couplings, etc., the symmetrical FGMs were introduced in order to satisfy the requirements. For material compositions distributed symmetrically across the graded direction, the rule of mixture was modified for evaluating the effective material properties of such materials with different mathematical models (Chi and Chung, 2006a, Chi and Chung, 2006b, Mahi et al., 2010):

2.2 Beam and plate theories

2.2.1 Beam theories

The wave equation describing the motion of a string was first established in the form of a differential equation by D’Alembert in his memoir published by the Berlin Academy in 1750. To consider dynamic response of engineering structures, the Euler-Bernoulli or classical beam theory (CBT), which has been widely used until recent years, was initially introduced by Euler in 1744 and Daniel Bernoulli in 1751 for dealing with the vibration problem of a thin beam supported by different end conditions (Rao, 2004). In the classical beam theory, the effects of rotary inertia and shear deformation were not taken into account. Therefore, Timoshenko (Timoshenko, 1921, Timoshenko, 1922, Wang, 1970)

Chapter 2

improved the beam theory, which has been well-known recently as the Timoshenko beam theory, covering the effects of rotary inertia and shear deformation. This theory is generally useful and appropriate for analyzing flexural problems of beams subjected to various mechanical loadings. Using the theory leads to much more precision in prediction beam behaviour than using the CBT, especially for the thick beams. However, the Timoshenko beam theory is one of the beam theories that need shear correction factors calculated from the beam cross-section. For example, for the beams having a rectangular cross-section, the use of shear correction factors varies from three over four to five over six. In order to represent the kinematics better without requiring shear correction factors which are normally needed in the first order shear deformation theory (FSDT), higher order shear deformation theories (HSDTs) were introduced and encouraged to describe beam behaviour under mechanical loadings. The third order shear deformation theory (TSDT) was developed among HSDTs that was suitable for composite material structures because it can yield more accurate interlaminar stress distributions. However, owing to the algebraic complexity and computational expense occurring with any higher order theory, the theories which have the order of shear deformation higher than the third order are not recommended. The displacement functions of TSDT are represented in the form of cubic variation along the thickness coordinate; consequently, the transverse shear strains and stresses can be obtained in the form of quadratic variation (Reddy, 1984, Levinson, 1981, Heyliger and Reddy, 1988, Reddy, 2004). Additionally, several types of classical and non- classical beam theories such as classical beam theory (CBT), first order shear deformation beam theory (FSDBT), parabolic shear deformation beam theory (PSDBT), hyperbolic shear deformation beam theory (HSDBT), exponential shear deformation beam theory (ESDBT), trigonometric shear deformation beam theory (TSDBT), Aydogdu shear deformation beam theory (ASDBT) were presented and discussed in (Aydogdu, 2005, Şimşek, 2010a). The general form of displacement field corresponding to various beam theories can be expressed as the following:

Chapter 2

(ݑሺݔǡ ݕǡ ݖǢ ݐሻ ൌݑ଴ሺݔǡ ݕǢ ݐሻ െݖݓǡ௫ ൅Ȱሺݖሻݑଵሺݔǡ ݕǢ ݐሻ, (2.9a (ݒሺݔǡ ݕǡ ݖǢ ݐሻ ൌͲ, (2.9b

(ݓሺݔǡ ݕǡ ݖǢ ݐሻ ൌݓ଴ሺݔǡ ݕǢ ݐሻ. (2.9c

To obtain the particular beam theory, the function of Ȱሺݖሻ with respect to considered beam theory is inserted into Eq. 2.18a. The lists of Ȱሺݖሻ for different beam theories are presented below:

(CBT: Ȱሺݖሻ ൌͲ (2.10 (FSDBT: Ȱሺݖሻ ൌݖ (2.11 (PSDBT: Ȱሺݖሻ ൌݖሺͳെͶݖଶΤ͵݄ଶሻ (2.12 (ሺͳΤ ʹሻ (2.13 HSDBT: Ȱሺݖሻ ൌ ݄ሺݖΤ ݄ሻ െ ݖ (ESDBT: Ȱሺݖሻ ൌݖሾെʹሺݖ݄ሻΤ ଶሿ (2.14 (TSDBT: Ȱሺݖሻ ൌ݄Τ ߨሺߨݖΤ ݄ሻ (2.15

మ (ASDBT: Ȱሺݖሻ ൌݖߙିଶሺ௭Τ ௛ሻ Ȁ୪୬ఈ with ߙ=3. (2.16

It is noted that ݑ଴ and ݓ଴ are middle surface displacement components along the x and z

axis. And ݑଵ is an unknown function that accounts for the effect of transverse shear strain on the beam at the middle surface.

The kinematics of deformation of a transverse normal using the common beam theories, (a) Bernoulli-Euler beam theory, (b) Timoshenko beam theory and (c) The third order shear deformation theory of Reddy-Bickford, are illustrated in Fig. 2.5. These theories have attracted the attention of engineers and researchers to implement analysis and design of beams.

Chapter 2

Fig. 2.5 Cross-section displacements of three different beam theories (Wang et al., 2000, Yesilce and Catal, 2010)

For Bernoulli-Euler beam theory, the deformation of such theory was assumed to satisfy the Kirchhoff hypothesis. The theory can provide sufficient accuracy when applying for thin beam analysis. The definition of thin beam is a beam which has relatively small thickness compared to its length; in general, if the length to thickness ratio of beam is more than 20, the beam is considered as a thin beam. Within the basics of the Kirchhoff hypothesis, the normal section to the un-deformed beam at middle plane remains normal after deformation to eliminate the transverse shear strain. Unlike the simplest theory of Bernoulli-Euler theory, the Timoshenko beam theory allows the cross-section displacements to rotate freely with the rotation of a normal to the axis of the beam function ׎ሺݔǡ ݖሻ; whereas in the Bernoulli-Euler theory, the rotation function is constrained by transverse displacement

using ݀ݓ଴Τ݀ݔ . By using the Timoshenko beam theory, the shear correction factor ሺߢሻ is 23

Chapter 2

required to be multiplied with the term of transverse shear force resultant. The common use of the factor is ߢൌͷΤ ͸ for isotropic materials. However, for structures made from FGMs, the shear correction factor depends on the Poisson’s ratio ሺߥሻ in which the equation used to obtain the factor is expressed as (Zhao et al., 2009a): ͷ൅ͷߥ (2.17) ߢሺߥሻ ൌ Ǥ ͸൅ͷߥ

The equation of the shear correction factor can be expressed in another form as (Efraim and Eisenberger, 2007, Zhao et al., 2009a): ͷ (2.18) ߢሺߥሻ ൌ ͸െሺߥଵܸଵ ൅ߥଶܸଶሻ where ߥଵand ߥଶare the Poisson’s ratios for ceramic and metal, ܸଵand ܸଶ denote the volume fractions for ceramic and metal across the entire cross-section.

The Timoshenko beam theory has some major numerical problems such as shear locking in the numerical analysis for composite materials (Yesilce and Catal, 2010). To overcome numerical and physical problems of beam analysis, higher order shear deformation theories (HSDTs) were developed. These theories relax the restriction on the warping of the cross- section and allow variation in the longitudinal direction of the beam with cubic variation. Reddy-Bickford beam theory (Wang et al., 2000) is one of the effective theories among HSDTs. The kinematics of deformation of a transverse normal based on the theory can be seen in Fig. 2.5(c). The Reddy-Bickford beam theory is more exact and provides better representation of the physics of the problem, which does not need the shear correction factor.

2.2.2 Layerwise theory for beam and plate analysis

The layerwise theory or 3-D elasticity theory which contains full 3-D kinematics and constitutive relations was developed in order to predict the mechanical behaviour of laminated composite components. The simple equivalent single-layer theories, for example, the classical and first order shear deformation theories are widely used for predicting the behaviour of thin laminated composite structures. These two traditional theories are often incapable of accurately determining the 3-D stress field at the composite layer. Due to 24

Chapter 2

several aspects such as thickness effects and the 3-D state of stress and strain at the composite layer being taken into account in the layerwise theory, the global response of composite components in terms of deflection, critical buckling loads and vibration frequencies can be obtained more precisely using the theory compared to those obtained from the two traditional ones.

The displacement field based on the layerwise theory can also give good kinematics of deformation as well as the representation of cross-sectional warping of thick composite components. Two displacement-based layerwise theories have been used widely: (1) the partial layerwise theory that refers to the layerwise expansions given for the in-plane displacement functions only whereas for the transverse displacement function is given the same as usual theories, (2) the full layerwise theory that refers to the layerwise expansions given for all of the displacement functions. An application of the layerwise theory accounts for the zigzag phenomenon of the in-plane displacements across the thickness of structures. By considering structural analysis of composite materials, the zigzag phenomenon could be pronounced when analyzing thick composite structures where the transverse shear modulus changes suddenly through the thickness. The total displacement field based on the layerwise theory can be written as:

ே (2.19a) ୍ ݑሺݔǡ ݕǡ ݖǡ ݐሻ ൌ෍ܷூሺݔǡ ݕǡ ݐሻȰ ሺሻ ூୀଵ ே (2.19b) ୍ ݒሺݔǡ ݕǡ ݖǡ ݐሻ ൌ෍ܸூሺݔǡ ݕǡ ݐሻȰ ሺሻ ூୀଵ ெ (2.19c) ୍ ݓሺݔǡ ݕǡ ݖǡ ݐሻ ൌ෍ܹூሺݔǡ ݕǡ ݐሻȲ ሺሻ ூୀଵ

where ሺܷூǡܸூǡܹூሻ are defined as the nodal values of ሺݑǡ ݒǡ ݓሻ, and N is the total number of nodes. Ȱூ is the global interpolation function for in-plane displacements throughout thickness. M and Ȳூ are given as the number of nodes and the global interpolation function of the transverse displacement along the thickness, respectively. The linear and quadratic

Chapter 2

variation of the global interpolation function for both in-plane and transverse displacements are introduced in Reddy (Reddy, 1992, Reddy, 2004).

Because of the layerwise theory demonstrating several benefits as presented above, many researchers have concentrated on employing the theory to investigate structural analysis of composite materials. Tahani (Tahani, 2007) used this theory to investigate static bending analysis and free vibration of laminated composite beams. The exact 3-D elasticity solution for the case of cross-ply laminates was adopted to compare with the layerwise solution. Bending and vibration results of angle-ply laminates which were obtained from the commercial finite element package of ANSYS were also used to confirm the accuracy of the theory. Additionally, for graded materials, bending and free vibration response of layered functionally graded beams using layerwise theory was investigated by Kapuria et al. (Kapuria et al., 2008a) including experimental validation. To solve composite plate problems with the layerwise theory, this was found in many research studies (Han and Lee, 1998, Ferreira et al., 2005, Shariyat, 2007, Plagianakos and Saravanos, 2009, Moleiro et al., 2010).

2.2.3 Common plate theories and an improved TSDT

By considering plate geometry, it is known that plate is an initially flat surface structure whose thickness is small in comparison with its other dimensions, while for beam geometry, the length is significantly greater than the depth and width. Therefore, beam analysis can be treated as a one-dimensional problem. In the case of plate analysis, the widths a and b that can be defined as the lengths along the x- and y-axis, respectively, are involved in analytical consideration. In such a case, the deflection (w) and displacements at a point on the mid-plane (u0,v0) of the plate are functions of x and y coordinates so that all derivatives with respect to x and y must be taken into account. Plate theory developments for composite materials in the past have been based on one of the following approaches (Reddy, 2004),

Chapter 2

(1) Equivalent single-layer theories (2-D) (a) Classical laminated plate theory (b) Shear deformation laminated plate theory (2) Three-dimensional elasticity theory (3-D) (a) Traditional 3-D elasticity formulations (b) Layerwise theories (3) Multiple model methods (2-D and 3-D)

Fundamentals and details of the classical, the first order shear deformation and the third order shear deformation plate theories were presented and demonstrated with a variety of mechanical problems (Reddy, 2004). The third order shear deformation theory (TSDT) developed by Reddy (Reddy, 2004) is one of the effective higher order theories that has been used widely for predicting plate analysis. The Reddy’s TSDT is based on the following displacement field:

(Ͷ ߲ݓ (2.20a ݑሺݔǡ ݕǡ ݖǡ ݐሻ ൌݑ ሺݔǡ ݕǡ ݐሻ ൅ݖ߶ ሺݔǡ ݕǡ ݐሻ െ ݖଷ ൬߶ ൅ ଴൰ ଴ ௫ ͵݄ଶ ௫ ߲ݔ

(Ͷ ߲ݓ (2.20b ݒሺݔǡ ݕǡ ݖǡ ݐሻ ൌݒ ሺݔǡ ݕǡ ݐሻ ൅ݖ߶ ሺݔǡ ݕǡ ݐሻ െ ݖଷ ൬߶ ൅ ଴൰ ଴ ௬ ͵݄ଶ ௬ ߲ݕ

(ݓሺݔǡ ݕǡ ݖǡ ݐሻ ൌݓ଴ሺݔǡ ݕǡ ݐሻ (2.20c

By using the displacement field presented in Eq. (2.20) to formulate normal strain ሺ߳ሻ and shear strain ሺߛሻ components, one can obtain the equations of strains presented in the form of lower and higher order components as follows.

Chapter 2

߲ݑ ۗ ଴ ۓ ሺ଴ሻ ߲ݔ ۖ ۖ ۗ ߳௫௫ۓ ߲ݒ଴ ߳ሺ଴ሻ ൌ (௬௬ ߲ݕ (2.21 ۘ ۔ ሺ଴ሻۘ ۔ ۖ ߛ௫௬ ۙ ۖ ߲ݑ ߲ݒە ൬ ଴ ൅ ଴൰ۖۖ ۙ ߲ݕ ߲ݔ ە

߲߶ ۗ ௫ ۓ ሺଵሻ ߲ݔ ۖ ۖ ۗ ߳௫௫ۓ ߲߶௬ ߳ሺଵሻ (௬௬ ൌ ߲ݕ (2.22 ۘ ۔ ሺଵሻۘ ۔ ۖ ߛ௫௬ ۙ ۖ ߲߶ ߲߶௬ە ቆ ௫ ൅ ቇۖۖ ۙ ߲ݕ ߲ݔ ە

߲߶ ߲ଶݓ ۗ ቆ ௫ ൅ ଴ቇ ۓ ሺଷሻ ߲ݔ ߲ݔଶ ۖ ۖ ߳ ۖ ௫௫ ۗ ۖ ଶ ۓ ߲߶௬ ߲ ݓ ߳ሺଷሻ ൌെܿ ቆ ൅ ଴ቇ (2.23) ௬௬ ଵ ߲ݕ ߲ݕଶ ۘ ۔ ሺଷሻۘ ۔ ߛ ଶ ۖ ௫௬ ۙ ۖ ߲߶ ߲߶௬ ߲ ݓ ە ቆ ௫ ൅ ൅ʹ ଴ቇۖۖ ۙ ߲ݔ ߲ݕ ߲ݔ߲ݕ ە

߲ݓ଴ ߲ݓ଴ ൬ ൅߶௬൰ۗۓ ൬ ൅߶௬൰ۗ ሺଶሻۓ ሺ଴ሻ ߛ௬௭ ߲ݕ ߛ௬௭ ߲ݕ ൝ ൡൌ ǡ൝ ൡൌെܿ (2.24) ሺ଴ሻ ሺଶሻ ଶ ۘ ߲ݓ଴ ۔ ۘ ߲ݓ଴ ۔ ߛ௫௭ ൬ ൅߶ ൰ ߛ௫௭ ൬ ൅߶ ൰ ۙ ߲ݔ ௫ ە ۙ ߲ݔ ௫ ە

ଶ where ܿଵ ൌͶΤ ͵݄ and ܿଶ ൌ͵ܿଵ. It is remarkable that the Reddy’s TSDT is simplified to the FSDT by setting ܿଵ ൌܿଶ ൌͲ. It is also specialized to the CPT by letting ܿଵ ൌܿଶ ൌ డ௪ డ௪ ߛሺ଴ሻ ൌߛሺ଴ሻ ൌߛሺଶሻ ൌߛሺଶሻͲ as well as replacing ߶ and ߶ by బ and బ. ௬௭ ௫௭ ௬௭ ௫௭ ௫ ௬ డ௫ డ௬

Aydogdu (Aydogdu, 2009) mentioned that plate deformation theories can be divided into two groups: stress based and displacement based theories. In his study, several higher order shear deformation theories according to displacement based theories were also presented. It is observed that the displacement field assumption of the plate theories can be used to treat beam analysis as well when the length along y-axis is very small compared to that along x- 28

Chapter 2

axis; therefore, all derivatives with respect to y are zero. In generality of plate theories, it is seen that in-plane displacement functions of u and v along the x and y coordinate directions are represented with similar form.

An improved third order shear deformation theory based on rigorous kinematics of displacements was developed by Shi (Shi, 2007). The theory was initially developed for static analysis of isotropic and orthotropic beams and plates. The author concluded that the improved theory provided more accuracy than other higher order shear deformable theories, especially when the transverse shear plays a very important role. Because of the kinematics of displacements in the improved TSDT derived from an elasticity formulation rather than the hypothesis of displacements, it is interesting to further implement this theory for the thermal buckling and free vibration analysis FG beams and plates, attempted in this research.

Beginning with the constitutive equations of such a plate which take the form as,

ߪ௫௫ ܳଵଵ ܳଵଶ Ͳ ߳௫௫ (2.25) ߬௬௭ ܳସସ Ͳ ߛ௬௭ ൝ߪ௬௬ൡൌ൥ܳଶଵ ܳଶଶ Ͳ ൩൝߳௬௬ൡǡቄ ቅൌ൤ ൨ቄ ቅǤ ߬௫௭ Ͳܳହହ ߛ௫௭ ߬௫௬ ͲͲܳ଺଺ ߛ௫௬

The displacement field of this theory is written as follows,

ܳ௫ Ͷ ଶ (2.26a) ݑൌݑ଴ሺݔǡ ݕǡ ݐሻ െݓ଴ǡ௫ݖ൅ ൬͵ݖ െ ଶ ݖ ൰ǡ ʹܳହହ݄ ݄

ܳ ௬ Ͷ ଶ (2.26b) ݒൌݑ଴ሺݔǡ ݕǡ ݐሻ െݓ଴ǡ௬ݖ൅ ൬͵ݖ െ ଶ ݖ ൰ǡ ʹܳସସ݄ ݄

(ݓൌݓ଴ሺݔǡ ݕǡ ݐሻǡ (2.26c ହ ହ .where ܳ ൌ ܳ ݄൫߶ ൅ݓ ൯ and ܳ ൌ ܳ ݄൫߶ ൅ݓ ൯ ௫ ଺ ହହ ௫ ଴ǡ௫ ௬ ଺ ସସ ௬ ଴ǡ௬

It is noted that ܳ௫ and ܳ௬ are the shear forces acting on the cross-sections with the normal

in the x and y coordinate directions; moreover, the physical meanings of ൫߶௫ ൅ݓ଴ǡ௫൯ and 29

Chapter 2

൫߶௬ ൅ݓ଴ǡ௬൯ terms are the transverse shears of the cross-section with x=constant and ହ y=constant, respectively. The factor is obtained from the work equivalence between the ଺ transverse shear forces and the transverse shear stresses.

2.2.4 The refined plate theory

An efficient and simple refined plate theory (RPT) was initially introduced and implemented by Shimpi and Patel (Shimpi and Patel, 2006a, Shimpi and Patel, 2006b) in order to deal with the problems of static and dynamic analysis of orthotropic plates. The refined theory can be classified among the third-order shear deformation theories. The development of the refined plate theory is based on the assumptions that the theory

represents parabolic variations of shear strains ( ߛ௫௭ǡߛ௬௭) and shear stresses (ߪ௫௭ǡߪ௬௭) throughout the plate thickness and also satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate. Additionally, the theory can provide high accuracy in prediction plate behaviour subjected to mechanical loadings without using the shear correction factor.

Owing to many advantages of using the RPT, the theory has recently attracted a lot of interest from researchers. Kim et al. (Kim et al., 2009a) used the RPT to solve the buckling problem of isotropic and orthotropic plates under various loadings such as uniaxial compression, biaxial compression and tensile-compressive combination in different directions. The authors have extended their work again using the RPT to deal with plate analysis in different problems. The RPT was applied to solve bending and vibration response of laminated composite plates. The closed-form solutions obtained by the Navier technique were presented to find out the deflection and natural frequency of the simply supported plates (Kim et al., 2009b, Thai and Kim, 2010). By using the Levy-type solution with the RPT to deal with buckling analysis of orthotropic plates, the buckling loads of the plates with two opposite edges simply supported and the other two edges supported by arbitrary boundary conditions were obtained (Thai and Kim, 2011).

Chapter 2

Based on the extensive literature survey, it is found that there are a few reports using the RPT to solve the problems of FG plates subjected to static and dynamic loadings. Thai and Choi (Thai and Choi, 2011) employed the RPT to analyse buckling problem of simply supported FG plates using Navier technique. For the FG plates with two opposite edges (at x=0 and x=a) simply supported and the other two edges (at y=0 and y=b) supported by simply supported, clamped and free conditions, the solutions of such plates can be achieved using the Levy-type solution. To investigate free vibration response of FG plates based on the RPT, Benachour et al. (Benachour et al., 2011) stated that a great benefit of using the RPT is that the fundamental frequencies of the FG plates can be solved accurately with small computational effort in comparison with other shear deformation theories. It is due to only four variables required for the RPT, as against five variables in the case of other shear deformation theories. Although some researchers have already used the RPT for analyzing laminated composite and FG plates, none of them has applied the theory to deal with thermal buckling and thermo-elastic vibration of FG plates.

Based on the basic assumptions of the RPT (Shimpi and Patel, 2006b, Kim et al., 2009a), the displacement field of the RPT can be written as follows:

ͳ ͷ (2.27a) ݑൌݑ ሺݔǡ ݕǡ ݐሻ െݖݓ ൅൬ ݖെ ݖଷ൰ݓ ଴ ௕ǡ௫ Ͷ ͵݄ଶ ௦ǡ௫

ͳ ͷ (2.27b) ݒൌݒ ሺݔǡ ݕǡ ݐሻ െݖݓ ൅൬ ݖെ ݖଷ൰ݓ ଴ ௕ǡ௬ Ͷ ͵݄ଶ ௦ǡ௬

(ݓൌݓ௕ሺݔǡ ݕǡ ݐሻ ൅ݓ௦ሺݔǡ ݕǡ ݐሻǤ (2.27c

The main differences between the improved TSDT developed by Shi (Shi, 2007) and the RPT are the middle terms of in-plane displacement functions. And the transverse displacement (w) of the RPT is composed of two components of the displacement due to

.(bending (ݓ௕) and shear (ݓ௦

Chapter 2

2.3 Bending analysis

In order to design structures made from FGMs subjected to mechanical loads under thermal environment, bending analysis is one of the necessary topics that needs to be considered. Bending causes problems of structural damage when the external bending moments applied to the structures exceeds their bending stiffness. FGMs are the new class of composite materials which are used to produce engineering structures. However, previous analyses of structures made by traditional materials such as isotropic and laminated composite materials are considered as a worthy background when designing FGM structures. Hence, in this research, some important researches dealing with bending, buckling and vibration analysis of isotropic and laminated composite structures are also reviewed.

2.3.1 Bending analysis of functionally graded beams

For bending analysis of isotropic and laminated composite beams, flexural analysis of fiber-reinforced composite beams subjected to transverse loading was investigated by Chandrashekhara and Bangera (Chandrashekhara and Bangera, 1993). In their investigation, the mathematical formulation based on higher-order shear deformation theory was derived for linear and geometrically non-linear analysis. The various parameters that have an effect on the non-linear deflection were chosen to study using the finite element method. Comparisons between theoretical and experimental results for bending and twisting of open-section composite beams were presented by Smith and Bank (Smith and Bank, 1992). To prepare for experimental investigation, symmetrically and anti- symmetrically laminated composite I-beams were fabricated by glass/polyester using hand- layup and vacuum bag techniques. According to the comparisons, excellent correlation among three sources (the experimental results, finite element results and analytical results) for the twist and bending deflection of the composite beams were observed. The layerwise theory was implemented by analysis of static bending and free vibration response of laminated beams in the study of Tahani (Tahani, 2007). The equations of motions based on the theory were obtained by using Hamilton’s principle and solved by analytical approach. Bending behaviour of a sandwich beam with a soft core and unsymmetrical laminated composite skins was chosen to investigate in study of (Frostig and Shenhar, 1995). The soft

Chapter 2

core was assumed to be compressible material. The significant effect of the extension- bending coupling due to unsymmetrical layups was also taken into account. Local effects in the vicinity of concentrated loadings that consist of dimples and high bending moments in the skins and sudden changes in the peeling stresses were investigated. A simple analytical method for calculating stresses and strains existing in thin-walled composite beams subjected to tension and bending loads was provided by Estivalezes and Barrau (Estivalezes and Barrau, 1998). Numerical results obtained from the simple method were brought to compare with the finite element software (MEF-MOSAIC) results.

In the case of bending analysis of functionally graded (FG) beams, Sankar (Sankar, 2001) provided an elasticity solution for calculating deflection, normal and shear stresses and bending moment of FG beams subjected to transverse loadings. The exponential law distribution was used to describe the changes of material properties throughout the beam thickness. The equations of motion of the FG beams were derived based on the Euler- Bernoulli beam theory. The expression of beam stiffness coefficients A, B and D can be defined as,

௛ (2.28) തሺͳǡ ݖǡ ݖଶሻ݀ݖܧ ሻ ൌනܦ ǡܤ ǡܣሺ ଴ ఒ௭ :ത଴݁ , the integral results can then be expressed asܧതሺݖሻ ൌܧ where

ܧത െܧത (2.29a) ܣൌ ௛ ଴ ߣ ݄ܧത െܣ (2.29b) ܤൌ ௛ ߣ ݄ଶܧത െʹܤ (2.29c) ܦൌ ௛ ߣ

ܧത଴ ൌܧതሺͲሻܧത௛ ൌܧതሺ݄ሻǤ (2.29d)

Zhong and Yu (Zhong and Yu, 2007) presented a general solution of a cantilever FG beam subjected to different forms of mechanical loads. A two-dimensional solution based on Airy stress function was formulated to solve the bending problem of such an FG beam with arbitrary variations of material properties. Kadoli et al. (Kadoli et al., 2008) used higher

Chapter 2

order shear deformation theory to solve the static problem of FG beams under ambient temperature. The static finite element equilibrium equations derived from the principle of stationary potential energy were employed to deal with metal-ceramic FG beams subjected to uniformly distributed load. According to the investigations, it was revealed that the power law index or the volume fraction index is the significant parameter, which affects on deflection, normal and shear stresses as well as the location of the neutral surface. Additionally, with the same loading magnitude, the distributed load acting on a metal surface did not provide similar results of slope or curvature of axial stress profile compared to those obtained from the load acting on ceramic surface. To understand the behaviour of an FG cantilever beam subjected to uniformly distributed load in thermal environment, Huang et al. (Huang et al., 2007) provided an analytical solution based on elementary formulations using plane stress assumption. Compatibility equation and boundary conditions were applied to determine the stress function before using to obtain the solutions of normal and shear stresses, bending moment, and deflection. Ying et al. (Ying et al., 2008) presented the exact solution of bending and free vibration analysis of FG beams resting on the Winkler-Pasternak elastic foundation. The important parameters such as gradient index, aspect ratios and foundation parameters were taken into consideration. In the investigation of Lu (Lu et al., 2007), material properties of FG beams were assumed to vary exponentially along the thickness and longitudinal directions. The bi-directional FG beams were chosen to consider bending analysis under thermal environment by using the state space-based differential quadrature method. According to the investigation, it was found that the reduction in thermal stresses can be achieved by selecting appropriate materials, whose properties against temperature loadings are small and distribute uniformly.

Warping and shear deformation effects of functionally graded short beams under three- point bending were found in the investigation of Benatta et al. (Benatta et al., 2008). In the study, high-order flexural theories were employed to formulate the governing equations based on the principle of virtual work. In some realistic situations, beams subjected to large amplitude loadings often exhibit material and geometrical non-linearity, Kang and Li (Kang and Li, 2009) considered the mechanical behaviour of a non-linear functionally graded material cantilever beam under an end force using small and large deformation theories.

Chapter 2

The non-linear stress-strain relationship according to the Ludwick type law was used to describe the stress component ሺߪ௫௫ሻ, which can be presented as follows:

ఒ (ሺݖଵሻȁߝ௫௫ȁ ǡͲ ൑ ݖଵ ൑݄ (2.30ܧߪ௫௫ ൌሺߝ௫௫ሻ where ߝ௫௫ is the strain component, ߣǡ a given material constant of material non-linearity and sign(*) denotes the symbol function. Zenkour et al. (Zenkour et al., 2009) gave bending analysis of FG viscoelastic sandwich beams with elastic cores. The faces of the sandwich beams were made of FG viscoelastic materials, while the core was made from elastic materials. The sandwich beams were assumed to place on the Pasternak’s elastic foundations.

2.3.2 Bending analysis of functionally graded plates By considering composite plate analysis, with the use of the classical analytical methods such as Navier and Levy-type solutions, the problems of plates subjected to static and dynamic loadings are solved effectively. Details and principles of those methods were presented and discussed in a useful text book of Reddy (Reddy, 2004). Exact solutions of thermal bending of graphite/epoxy and glass/resin composite plates were presented in Ref. (Kameswara Rao, 1984). The influence of several aspects covered in this study can be seen as the volume fraction of fibres, boundary conditions and plate aspect ratios.

A simply supported FG plate was conducted to consider nonlinear bending analysis under thermal environment in the investigation of Shen (Shen, 2002). The FG plate was subjected to transverse uniform and sinusoidal loadings. The Reddy’s higher order shear deformation plate theory was employed to formulate the governing equations of such plate, and a mixed Galerkin-perturbation technique was used to solve the governing equations in order to obtain load-deflection and load-bending moment curves. For thermo-mechanical loads acting on the FG plates in the study of Yang and Shen (Yang and Shen, 2003b), the plates that were clamped or simply supported on two opposite edges and the remaining edges may be supported by arbitrary conditions were studied using large deflection analysis. By considering cylindrical bending analysis of the FG plates including thermal effect, Qian et al. (Qian et al., 2004) used a meshless local Petrov-Galerkin method to solve the problem.

Chapter 2

The effective material properties of the FG plates in this investigation were assumed to obey the Mori-Tanaka model. Zenkour (Zenkour, 2006) introduced the generalized shear deformation theory to deal with bending analysis of FG plates under transverse uniform loading. In the generalized theory, it was simplified by enforcing traction-free boundary conditions at the plate faces. Several effects, for example transversal shear deformation, plate aspect ratio, side-to-thickness ratio and material volume fraction, that lead to considerable changes in transverse deflections of the FG plates were taken into consideration. Na and Kim (Na and Kim, 2006a) applied the Green-Lagrange nonlinear strain displacement to solve bending response of FG plates under uniform pressure and thermal loads. The assumptions of temperature rise across the plate thickness were set as uniform, linear and sinusoidal variation. To solve the problem, a 3-D finite element method was adopted to determine the numerical results. By using the uniform temperature rise, it was found that the largest value of deflection can be obtained, whereas, the sinusoidal temperature rise gives the smallest one. Classical plate theory (CPT) with the von Karman strains was applied to produce the nonlinear equilibrium equations for analyzing the FG plates under in-plane and transverse loadings (Navazi et al., 2006). The combination between thermal and electrical loads acting on a simply supported FG plate with or without surface bonded piezoelectric actuators was considered in the study of Shen (Shen, 2007a). By applying thermal loadings in this study, the heat conduction and temperature-dependent material properties were also involved in calculations. Based on the investigating results, it was revealed that temperature-dependency of FGMs could be pronounced for considering thermal bending analysis of such plate. However, the control voltage provided little impact on the thermal bending response of the plate. Brischetto et al. (Brischetto et al., 2008) extended the unified formulation (UF), which has been successfully used for multi-layered structures, to deal with thermo-mechanical bending of FG plates. In order to neglect the stretching-bending coupling stiffness component in FGMs, a physical neutral surface would be given as the reference plane. The position of the surface (at z = z0) can be obtained by the following equation:

Chapter 2

௛ ଶ ሺݖሻ݀ݖܧ׬ ௛ ݖ ି ଶ ݖ଴ ൌ ௛ Ǥ ଶ (2.31) ሺݖሻ݀ݖܧ ׬ ௛ ି ଶ

It is observed that the physical neutral and geometric middle surface are the same when considering homogeneous isotropic and symmetrical laminated composite materials. Zhang and Zhou (Zhang and Zhou, 2008) used the physical neutral surface instead of the geometric middle surface to analyse bending, buckling and vibration response of FGM thin plates. Next, an exact solution was applied to solve a fourth-order differential equation of nonlinear cylindrical bending analysis of FG plates. The material properties of FG plates were assumed to vary according to the simple power law function and Mori-Tanaka scheme. The deflection results due to in-plane and transverse loadings were presented and discussed for a case of simply supported boundary condition (Navazi and Haddadpour, 2008). The layerwise theory was employed to investigate deflection and stresses in FG plates under thermo-mechanical loadings. To solve the equilibrium equations based on the layerwise theory, a perturbation technique was implemented to obtain the numerical results (Tahani and Mirzababaee, 2009). Zenkour (Zenkour, 2009) applied the refined sinusoidal shear deformation plate theory to analyse the bending problem of FG plates resting on the Pasternak elastic foundation. Comparisons between the results of deflection and stresses obtained from the sinusoidal theory and those obtained from other plate theories were demonstrated in this investigation. A simply supported FG plate resting on an elastic foundation was considered again by Shen and Wang (Shen and Wang, 2010) under transverse uniform or sinusoidal load combined with initial compressive edge loads. Two steps of perturbation technique were implemented to solve the nonlinear bending response of FG plate in order to carry out the load-deflection and load-bending moment relationship. Apart from the rectangular FG plates, other plate geometries, for example sectorial, annular and circular plates made of FGMs, were also considered for bending analysis, which were found in many reports (Reddy et al., 1999, Jomehzadeh et al., 2009, Sahraee, 2009).

Chapter 2

2.4 Stability analysis

Static instability of structures or buckling analysis of structures was initially introduced by Euler more than two centuries ago. The in-plane loads, represented in the form of mechanical and thermal loads, are known to be the main causes of buckling in structures. Objectives of buckling analysis are to determine the buckling critical loads and their corresponding mode shapes. In general, the transverse displacement (w) is more evident after the critical load is achieved. Therefore, the bifurcation point is defined at the critical load. The structure exhibits buckled configuration when subjected to the in-plane load that exceeds its critical load. Hence, post buckling analysis could be addressed to investigate structural behaviour at the stage of instability.

The basic concepts of elastic stability and closed form and approximated solutions for solving stability problems of structural elements subjected to external in-plane loadings, were provided and discussed in Ref (Wang et al., 2005, Simitses and Hodges, 2006). Several fundamentals of stability analysis such as buckling of straight, curved beams, lateral-torsional buckling of beams, dynamic stability, non-conservative systems etc. were also demonstrated in these references.

2.4.1 Stability analysis of functionally graded beams

Birman (Birman, 1990) studied buckling and bending of non-uniformly heated beams using a simple approach. The material properties of the heated beams were assumed to remain unaffected by variations of temperature. For composite beams, symmetric and non- symmetric cross-ply composite laminated beams made by Kevlar-epoxy, carbon-epoxy and glass-epoxy were chosen to investigate their thermal buckling behaviour using the first- order shear deformation (Abramovich, 1994). A laminated composite beam with embedded shape memory alloy (SMA), the smart material for enhancing the critical buckling temperature and reducing the lateral deflection, was used to study thermal buckling and post-buckling analysis (Lee and Choi, 1999). Exact analytical solutions of thermal buckling analysis of laminated composite beams were provided by Khdeir (Khdeir, 2001) using classical and shear deformation theories. Similarly, Ayadogdu (Aydogdu, 2007) used three

Chapter 2

different theories as seen with parabolic, hyperbolic and exponential shear deformation theories to analyse thermal buckling of cross-ply laminated composite beams with various boundary conditions. By using the Rayleigh-Ritz method with the admissible displacement functions based on the concept of coupled displacement field (CDF) criteria, thermal post- buckling results of slender columns associated with several axially immovable end conditions can be obtained easily with excellent accuracy (Gupta et al., 2010).

For stability analyses of functionally graded (FG) beams, Shi-rong et al. (Shi-rong et al., 2006) used a shooting method to solve problems of thermal post-buckling of fixed-fixed FG beams under non-uniform temperature rise. The governing equations were created according to Timoshenko beam theory for investigating behaviour of the FG beams at the unstable region. To further investigate this subject, non-uniform columns made of FGMs in which their material properties were set to change in axial direction were chosen to consider mechanical buckling analysis with several end conditions (Singh and Li, 2009). Within the framework of Timoshenko beam theory, Ke at al. (Ke et al., 2009b) presented post-buckling analysis due to in-plane loading of edge cracked FG beams with several impacts of material composition, crack depth, crack location and slenderness ratio. Thermal buckling behaviour of FG beams associated with different boundary conditions were investigated by Kiani and Eslami (Kiani and Eslami, 2010) using the Euler-Bernoulli beam theory. Three types of thermal loads throughout the beam thickness; namely, uniform, linear and non-linear temperature rise, were taken into consideration. Based on the numerical results, it was revealed that the lowest critical temperature of such beams for every thickness ratio can be obtained from the uniformly thermal load and followed with linear and non-linear thermal loads, respectively. Next, the comparisons between the classical Rayleigh-Ritz (R-R) method and the versatile Finite Element Analysis (FEA) were demonstrated to study thermal post-buckling analysis of uniform slender FG beams using the classical beam theory. By using one term approximation for the R-R method, the closed form solutions of nonlinear critical loading of simply supported and clamped FG beams can be obtained easily with good accuracy when compared to the FEA results (Anandrao et al., 2010).

Chapter 2

2.4.2 Stability analysis of functionally graded plates

In the past, buckling analysis of isotropic rectangular plates due to mechanical and thermal loadings had been reported in many studies (Chen et al., 1982, Gauss and Antman, 1984, Ascione, 1981, Bednarczyk and Richter, 1985, Xiang-Sheng, 1981). To understand the buckling behaviour of arbitrarily laminated composite plates subjected to in-plane loadings, Sharma et al. (Sharma et al., 1980) presented the closed form solution in order to determine the buckling response of anti-symmetric cross-and angle-ply plates for various boundary conditions. The effect of coupling between bending and extension that occurs with antisymmetric composites gives a quite substantial buckling load. Classical bifurcation buckling analysis of composite plates was discussed by Leissa (Leissa, 1983) and other important parameters such as elastic foundation, variable thickness, shear deformation, hygrothermal effects as well as in-plane heterogeneity were also included in this study. By using the classical plate theory (CPT) to analyse buckling of composite plates, many reports, as seen in open literature (Chai and Hoon, 1992, Lagace et al., 1986, Hu et al., 2003), were found to have analysed buckling response of such plates by using the CPT. Bruno and Lato (Bruno and Lato, 1991) gave the finite element solution to buckling analysis of moderately thick composite plates associated with a shear deformable plate model. To predict more precisely the results of critical load of laminated composite plates, several theories that cover shear deformation effect were introduced to investigate buckling behaviour of the plates (Argyris and Tenek, 1994, Gilat et al., 2001, Timarci and Aydogdu, 2005, Aydogdu, 2009). For thermal buckling of laminated composite plates, this topic was studied to find out the thermal buckling parameter using different methods (Chen and Chen, 1987, Chen and Chen, 1989, Raju and Rao, 1989). Analytical three-dimensional elasticity solutions were also given by Noor and Burton (Noor and Burton, 1992) for thermal buckling of multilayered anisotropic plates. The effect of initial geometrical imperfection on thermal buckling of composite plates was first investigated by Shen and Lin (Shen and Lin, 1995) using a mixed Galerkin-perturbation technique. To predict more precisely analytical results, the Reddy’s higher order shear deformation theory was used to analyse thermal post-buckling of composite plates under uniform and non-uniform temperature loadings. Several important effects such as transverse shear deformation, thermal load ratio,

Chapter 2

fiber orientation etc. were conducted to investigate with the composite plates that have an initial geometrical imperfection. All of the effects, considered in this investigation, had shown significant impacts on thermal buckling results except for the effect of the total number of plies (Shen, 1997).

Equilibrium and stability equations of functionally graded (FG) plates made of alumina

(Al2O3) and aluminum (Al) under three different thermal loads; namely, uniform, linear and non-linear temperature rise were investigated using classical and higher order theory. This investigation revealed several important outcomes for FGM structural design, for example, by considering non-linear temperature distribution across the thickness of the plate, the greatest buckling temperature can be obtained in comparison with other types of temperature distribution. The aspect and thickness ratios produced significant impacts on the buckling temperature results. Using the classical plate theory, buckling temperature results are generally over-estimated as compared to those obtained from the higher order theory (Javaheri and Eslami, 2002a, Javaheri and Eslami, 2002b). In the study of Chen and Liew (Chen and Liew, 2004), the mesh-free method based on the radial basis function was applied to the problem of buckling analysis of FG plates subjected to pin loads, partial uniform loads and parabolic loads. On the basic of FSDT, Lanhe (Lanhe, 2004) gave the solutions for thermal buckling analysis of simply supported moderate and thick FG plates. The influences of aspect ratio, the relative thickness, the gradient index as well as the transverse shear that lead to considerable changes in buckling temperature results were also investigated and discussed. From this investigation, it can be concluded that the critical buckling temperature of FG plates is generally lower than the corresponding values for homogenous plates, therefore, it is important for design to check the strength of FG plates subjected to thermal load. For three-dimensional analysis of thermal buckling of FG plates, the eigenvalue problem of such plates can be solved by the finite element method using 18- node solid element (Na and Kim, 2004, Na and Kim, 2006b, Na and Kim, 2006c). The influence of geometrical imperfection on thermal instability of FG plates was investigated by Samsamshariat and Eslami (Samsamshariat and Eslami, 2006) using the classical plate theory. The imperfections of general plate can be assumed as follows:

Chapter 2

ߨݔ ݊ߨݕ݉ (ൌߤ݄ ǡ݉ǡ ݊ ൌ ͳǡʹǤǤ (2.32 כݓ ܽ ܾ where the coefficient ߤ varies from 0 to 1, and ߤ݄ represents the imperfection size. It is also defined that m and n are the number of half waves in x and y coordinates, respectively. According to the effect of the imperfection, it can be concluded that the buckling temperature of the imperfect FG plate is greater than its counterpart. Increasing imperfection size leads to an increase of the buckling temperature.

Buckling analysis of FG plates under mechanical and thermal loadings was considered by using TSDT (Shariat and Eslami, 2007). Thermal post-buckling analysis of simply supported FG plates based on TSDT was studied by Shen (Shen, 2007b) using two cases of temperature field, i.e. in-plane non-uniform parabolic temperature distribution and heat conduction. It was found that, in the case of heat conduction, the post-buckling path of geometrically perfect plates is no longer of the bifurcation type. By using fast converging finite double Chebyshev polynomials, the post-buckling response of FG plates due to mechanical and thermal loadings was studied and presented by Wu et al. (Wu et al., 2007). It was stated in the study that the solution methodology could be extended for finding the post-buckling response of hybrid FG plates subjected to mixed loading conditions. Zhao et al. (Zhao et al., 2009b) used FSDT to solve the problems of mechanical and thermal buckling of FG plates using the element-free kp-Ritz method. To avoid the difficulty of shear locking effect for very thin plates in this study, the bending stiffness was evaluated using a stabilized conforming nodal integration technique. The shear and membrane terms were calculated using a direct nodal integration method. The significant part covered in this study (Zhao et al., 2009b) was to investigate the FG plates that contain square and circular holes at the centre. Consequently, it was found that the size of the hole presents a considerable impact not only on the buckling loading but also on the buckling mode shapes of the plate. Matsunaga (Matsunaga, 2009) applied a 2D higher-order deformation theory to find out thermal buckling results of FG plates, and concluded that by using such theory, accurate results could be obtained. A simple analytical approach was provided to investigate the stability of FG plates due to in-plane compressive, thermal and combined loads. These results illustrated that post-buckling behaviour of FG plates was greatly

Chapter 2

affected by material, geometric parameters, and in-plane boundary condition as well as initial imperfections (Tung and Duc, 2010). Three approximated analytical solutions, (a) Bubnov-Galerkin solution (BGS), (b) Power series solution (PSS) and (c) Semi-Levy solution (SLS) were applied to obtain thermal buckling results of clamped thin rectangular FG plates resting on Pasternak elastic foundation. From this study, it was found that increasing the Winkler constant or shear constant of the elastic foundation led to an increase in critical temperature. The buckling temperature was also affected by the effects of the shear interactions of the vertical elements; while, ΔTcr increased piecewise linearly with the Winkler foundation stiffness (Kiani et al., 2011).

For thermal buckling analysis of other plate geometries such as circular, sector, skew plates made from functionally graded materials, the thermal stability response of such plates was the main focus of many researchers using different theories and methodologies (Li et al., 2007, Prakash et al., 2008, Saidi and Hasani Baferani, 2010).

From the open literature, it was found that buckling analyses due to thermal loads of FG beams and plates were relatively rare and most of the past studies were simplified by assuming that the material properties remain constant within a certain range of temperature. In contrast, for composite materials and numerous alloys, this simplification was unacceptable when these materials were subjected to variations of temperature of several hundred degrees (Birman, 1990). As a result, in the current research work, the solutions of thermal buckling analysis of FG structures are derived by using temperature dependent material properties.

2.5 Vibration analysis

When engineering structures are subjected to dynamic loads, it is important to consider their vibration behaviour in order to design the structures appropriately. Natural frequencies of structural systems are the significant aspects that can be determined using different theories and methodologies. To avoid the resonance situation which may lead to structural damages due to dynamic loads, the structural system should be well-designed to have its natural frequencies which do not correspond to the frequencies of the dynamic loads acting 43

Chapter 2

on the system. Due to the significances of vibration analysis for structural design, the explosion of interest in vibration investigation has increased considerably in the past decades.

2.5.1 Vibration analysis of functionally graded beams

Since the development of fibre-reinforced composite materials for the purpose of making high strength to weight ratio materials, the investigation on the vibration response of beams made by the composite materials have drawn researchers’ attention. For example, many researchers (Teoh and Huang, 1977, Chandrashekhara et al., 1990, Abramovich, 1992, Chen et al., 2004, Aydogdu, 2005) studied vibration analysis of fibre-reinforced composite beams with the effects of shear deformation and fibre orientation using different theories and solution methods. By including thermal effect on vibration behaviour of the composite beams, this subject was found in several previous investigations (Ko and Kim, 2003, Fallah et al., 2011). For large amplitude vibration analysis of composite beams, Pai and Nayfeh (Pai and Nayfeh, 1990) used Newton’s second law to formulate the nonlinear equations to analyse the 3-D nonlinear vibration of composite beams. Based on von Karman’s large deflection theory, free vibration of unsymmetrically laminated beams was presented with various influences (Singh et al., 1991). Furthermore, composite beams resting on an elastic foundation with constrained ends were chosen to investigate their large amplitude vibration using DQM (Malekzadeh and Vosoughi, 2009).

For vibration characteristics of FG beams, the problems of static, free vibration and wave propagation of FG beams were solved using the effective element formulation (Chakraborty et al., 2003). The new beam element in this study was constructed by interpolating polynomials, which was related to the exact solution of the governing differential equations of the static part. Using the new element formulation, the stiffness matrix has super-convergent property and the element is free of shear locking. A simple solution was used to obtain the frequency parameter of a simply supported FG beam in the study of Aydogdu and Taskin (Aydogdu and Taskin, 2007). Material properties of the FG beam were assumed to vary according to power law and exponential law. Governing

Chapter 2

equations were formulated with three different beam theories, i.e. classical beam theory, parabolic and exponential shear deformation beam theories. The research outcomes revealed that frequency results obtained from the classical beam theory were over-predicted in comparison with those of other theories. Kapuria et al. (Kapuria et al., 2008a, Kapuria et al., 2008b) had made Al/SiC and Ni/Al2O3 beams using powder metallurgy and thermal spraying techniques in order to study the behaviour of static and free vibration response. Young’s modulus of the two beam systems was predicted by the modified rule of mixtures (MROM) which involves a transfer between stress-strain parameter in two phases of materials. As a result, the predictions using the MROM led to a close agreement with the experiment for both systems.

Free and forced vibration response of laminated functionally graded beams in which their

thickness ݄௦ሺݔሻ were set to vary as the function of ݄௦ሺݔሻ ൌ݄଴ሾͳ൅ߤ଴ߤሺݔሻሿ, where ݄଴ is -the reference thickness and ߤሺݔሻ is the thickness variation along the x-axis. The non uniformity, ߤ଴ ്Ͳ, represents a beam with variable thickness. It was found that the beam thickness variation is one of the significant impacts on both the free vibration and dynamic response (Xiang and Yang, 2008). Using Bernoulli-Euler beam theory and the rotational spring model, free vibration and buckling response of FG beams with edge cracks were considered by Yang and Chen (Yang and Chen, 2008). The flexibility of the beam G owing to the presence of the crack can be expressed according to Broek’s approximation (Broek, 1986);

ͳെߤଶ ܯଶ ܩ (2.33) ܭ ൌ ூ ܽ ʹ ሺݖሻ ூܧ where ܯூ is the bending moment at the cracked section, ܭூ is the stress intensity factor (SIF) under mode I loading. Therefore, crack location and total number of cracks were considered as the major features that can affect buckling and frequency results of the FG beams. A similar subject was also presented by other authors (Yu and Chu, 2009, Kitipornchai et al., 2009, Matbuly et al., 2009) using different theories and solution methods.

Chapter 2

Sina et al. (Sina et al., 2009) presented a new beam theory based on FSDT and compared its frequency results to those derived from the traditional one. In general, for the traditional FSDT, the stress and moment resultants in the lateral direction are assumed to be zero. Whereas, for the new FSDT, some terms of the resultants still exist (see (Sina et al., 2009)) for the following derivatives. In the case of using different higher order beam theories for vibration analysis of FG beams, the comparisons of fundamental frequencies obtained from the different theories were found in the literature (Şimşek, 2010a). It is observed that the frequency results of the higher order theories are normally very close to each other.

Free and forced vibration characteristics of FG beams under a concentrated moving harmonic load was considered using the Euler-Bernoulli beam theory. The influences of velocity of the moving harmonic load and the excitation frequency on the dynamic responses of the beams, which are the important aspects related to the characteristics of such beams, were investigated and discussed. According to the investigation results, it was found that the changes of the velocity of the moving load led to substantial changes of deflection of FG beams for every volume fraction index (Şimşek and Kocatürk, 2009). Şimşek (Şimşek, 2010c) extended the research work to deal with vibration of FG beams under a moving mass. The parametric studies covered within this work were the effects of shear deformation, material distribution, velocity of the moving mass, the inertia, Coriolis and the centripetal effects of the moving mass on the dynamic displacements and the stresses of the FG beams. Huang and Li (Huang and Li, 2010) introduced a simply novel approach to deal with free vibration of axially functionally graded beams with non-uniform cross-section. The principle of the new approach is to transform the governing differential equation with variable coefficients associated with appropriate boundary conditions to Fredholm integral equations. The study was based not only on the introduction of the new approach but also on the investigation of geometrical and gradient parameters. For symmetric FG beams, Mahi et al. (Mahi et al., 2010) presented the exact solution to study the free vibration response of such beams. Material properties with temperature dependency were assumed to obey the power law, exponential law and sigmoid law distribution. In terms of temperature distribution across the beam thickness, 1-D steady-state heat conduction equation (Fourier’s differential equation) used in their study is as follows:

Chapter 2

݀ ݀ܶ (2.34) ൤݇ሺݖሻ ൨ൌͲǡ ݖ ݀ݖ݀

݄ ݄ (2.35) ൬ݖൌ ൰ൌܶǡܶሺݖൌͲሻ ൌܶ ǡܶ ൬ݖ ൌ െ ൰ൌܶܶ ʹ ௧ ௠௣ ʹ ௕ where ܶ௧, ܶ௕, ܶ௠௣ are the temperature at the top, the bottom and the mid-plane surface of the beam, respectively. k (z) is the coefficient of thermal conductivity. The influences of temperature, material distribution based on different laws and the beam aspect ratio that yield the significant impacts on natural frequencies were investigated and discussed.

Structures which are subjected to a severe dynamic loading always exhibit large vibration amplitudes. To study the vibration behaviour of FG beams with large amplitude, the nonlinear analysis is required to deal with the problem. Ke et al. (Ke et al., 2009a) used the Euler-Bernoulli beam theory with the von Karman type nonlinear strain-displacement relationship that can be seen as follows:

߲ܷ ߲ଶܹ ͳ ߲ܹ ଶ (2.36) ߳ ൌ െݖ ൅ ൬ ൰ ௫ ߲ݔ ߲ݔଶ ʹ ߲ݔ

The nonlinear analysis governing differential equations based on the theory were solved by direct numerical integration method and Runge-Kutta method. Numerical results revealed that, for all types of FG beams, typical hardening behaviour is exhibited. Additionally, the nonlinear frequencies of several types of boundary conditions for FG beams are dependent on the sign of the vibration amplitude except for the case of a clamped-clamped FG beam that shows the same behaviour as homogenous beams. By using the Timoshenko beam theory for the nonlinear vibration of FG beams, Şimşek (Şimşek, 2010b) provided the solution based on the aid of the Newmark-β method in conjunction with the direct iteration method. Therefore, the problem of nonlinear vibration of FG beams under the action of a moving harmonic load can be solved. To further consider nonlinear vibration analysis and post-buckling response of FG beams, Fallah and Aghdam (Fallah and Aghdam, 2011) included the effects of nonlinear elastic foundation to study the phenomena using He’s variational method. The results of frequency and buckling load ratio and the proportion of

Chapter 2

linear and nonlinear results, were presented and discussed corresponding to elastic foundation coefficients. It was found that the frequency and buckling load ratio increase together with the increment of the coefficient values.

2.5.2 Vibration analysis of functionally graded plates

It was found that studies on the vibration response of laminated composite plates began to appear in the open literature many decades ago. Some early examples of researches concerning vibration analysis of laminated composite plates were found in several investigations (Elman and Knoell, 1971, Whitney, 1972, Noor, 1973). The area of vibration analysis of laminated composite plates has been continuously investigated to the current stage, although the studies began in the past decades. Furthermore, many substantial effects of nonlinearity, elastic constrained boundary condition, shear deformation, geometrical imperfection, temperature changes, damping, etc. that considerably affect the frequency results were also included in many studies (Chia, 1985, Chai, 1996, Liu and Huang, 1996, Raja et al., 2004, Nguyenvan et al., 2008, Ferreira et al., 2009, Naserian-Nik and Tahani, 2010).

Analyses of vibration characteristics of FG plates have been an interesting field of study to many researchers in the last decade. Due to FGMs consisting of ceramic and metallic phases in the system, therefore, the mechanical and thermo-mechanic performances of the system can be improved significantly in order to withstand problems when the materials are exposed to extreme temperature. In general applications of FG plates, the structures are used for many engineering applications under thermal environments. As a result, in several previous studies dealing with vibration analysis of FG plates, the thermal effect was one of the important effects that needed to be highlighted. To control the percentage of ceramics or metals in FGMs, the gradient index is used to describe the material constituents so that the index is also a significant parameter. Changing the gradient index means changing flexibility and density of FG structures.

Chapter 2

Free and forced vibration analysis of FG plates, which had initial stresses generated by thermal forces, was considered by Yang and Shen (Yang and Shen, 2001, Yang and Shen, 2002). Theoretical formulations based on classical plate theory and Reddy’s higher order shear deformation theory were constructed in the studies to find out the natural frequency and deflection results. The FG plates considered in the studies were assumed to be clamped on two opposite edges with the remaining edges either simply supported, free or clamped. The combined method of one dimensional differential quadrature, Galerkin and the modal superposition technique was applied to determine the transient response of the FG plates under lateral dynamic loads. According to the investigated results, it was found that, when thermal effect is included, FG plates with material properties intermediate to those of isotropic ones do not give intermediate frequency results and dynamic responses. In the study of Kim (Kim, 2005), the frequency results of FG plates with all edges clamped were illustrated using TSDT. The Rayleigh-Ritz procedure with the displacement functions in the form of the double Fourier series was applied to determine the temperature dependent vibration results. Two types of temperature field on the behaviour of the FGM were selected to be analysed i.e. thermal condition I and II.

Thermal condition I:

By imposing boundary condition of ܶሺݖሻ ൌܶ଴ ൅οܶ௎ at the upper surface (ݖൌ݄Τ ʹ) and

ሺݖሻ ൌܶ଴ ൅οܶ௅ at the lower surface (ݖൌ݄Τ ʹ), one can obtain the solution of steady-stateܶ heat transfer equation through the plate thickness. The solution can be expressed as follows:

(ሺݖሻ ൌܶ଴ ൅οܶሺݖሻ (2.37ܶ where

οܶ െοܶ ௭ ͳ (2.38) οܶሺݖሻ ൌοܶ ൅ ௎ ௅ න ݀ݖǤ ௅ ௛ଶΤ ͳ ݇ሺݖሻ ׬ ݀ݖ ି௛Τ ଶ ௛Τ ଶ ݇ሺݖሻି

Thermal condition II:

It was assumed that the low surface was given as ܶ௅ ൌܶ଴ ൅οܶ௅ , and the heat flow from ௐ the upper to the lower surface was ݍሺ ሻ. Notice that the heat transfer rate per unit area (q= ௠మ

Chapter 2

ௗ்ሺ௭ሻ ൌെ݇ሺݖሻ Ǥ Hence, theݍ heat flux) can be obtained as the following equation ௗ௭ temperature rise through the thickness can be represented in the form:

ͳ (2.39) න ݀ݖǤݍοܶሺݖሻ ൌοܶ ൅ ௅ ݇ሺݖሻ

To apply the nonlinear finite element method based on FSDT, Park and Kim (Park and Kim, 2006) presented the numerical results of the characteristics of the thermal post- buckling and vibration of FG plates in the pre- and post-buckled regions. Based on the numerical results, it can be concluded that the behaviour of FG plates in terms of thermal post-buckling and vibration are different from those of isotropic plates, and the volume fraction index shows a considerable impact on the results. However, in a further investigation on the behaviour of FG plates, Abrate (Abrate, 2006, Abrate, 2008) stated that FG plates behave like homogeneous plates if an appropriate reference plane is selected. Hence, the equations of motion are uncoupled and free of the effects of coupling between the in-plane and bending deformation. Consequently, the numerical results of bending, buckling and vibration can be defined and obtained with no need of any special technique or software. The idea of using the appropriate reference plane for the FG plates was considered again in the study of Zhang and Zhou (Zhang and Zhou, 2008). The appropriate plane in this study was defined as the physical neutral surface in which its position ሺݖ ൌ

ݖ଴ሻ can be obtained from Eq. (2.31). It is noted that for homogeneous isotropic and symmetrical laminated plates, the physical neutral surface and geometric middle surface are the same. In the present study the full membrane-extensional coupling terms are included so that the geometric middle surface is used as the original of z axis.

Natural frequencies and buckling stresses of simply supported FG plates were predicted accurately using a 2-D higher-order deformation theory. Higher shear deformation which has impacts on the natural frequencies and buckling stresses of the FG plates was presented for arbitrary values of the volume fraction index (Matsunaga, 2008). By applying the element-free kp-Ritz method, free vibration problem of square and skew FG plates with various boundary conditions was solved effectively to provide accurate frequency results

Chapter 2

with computational stability. This proposed method was used to find out the solution based on the FSDT so that the effects of transverse shear deformation and rotary inertia were also included. According to the results, it is found that, in the case of the FG plates with ݄ܽ൒ͳͲΤ , the shear correction coefficient is no longer important (Zhao et al., 2009a). To solve the free vibration problem of FG rectangular plates placed on either Winkler or Pasternak elastic foundations, the exact solutions to deal with the problem were presented and discussed in detail (Hosseini-Hashemi et al., 2010). In contrast, Liu et al. (Liu et al., 2010) proposed a different type of FG plates in which their material properties are assumed to vary in the direction of in-plane coordinates rather than the thickness direction like the common FG plates. The FG plates that have in-plane material inhomogeneity were considered to investigate their free vibration phenomena.

In terms of three-dimensional (3-D) vibration analysis of thick FG plates, the solutions were provided in several investigations. For example, Malekzadeh (Malekzadeh, 2009) studied the vibration analysis of FG plates resting on an elastic foundation with various boundary conditions. To solve the equations of motion associated with the 3-D constitutive relations, the differential quadrature method (DQM) was applied to obtain the results. By using another solution method, Li et al. (Li et al., 2009) applied the Ritz method with the assumed displacement function based on a series of Chebyshev polynomials to solve the 3- D vibration analysis for FG plates in a thermal environment. With moving line loads acting on the FG plate strips, the 3-D vibration analysis of such plates was found in the study of Hasheminejad and Rafsanjani (Hasheminejad and Rafsanjani, 2009).

By using the finite element method to address the problem of nonlinear free vibration analysis of rectangular and skew FG plates, the variation of nonlinear frequency ratio of such plates was obtained with several parametric studies such as gradient index, temperature change, skew angle, etc. (Sundararajan et al., 2005).

Considering vibration analysis of non-rectangular FG plates, Allahverdizadeh et al. (Allahverdizadeh et al., 2007b) provided the nonlinear vibration solution of FG circular plates using a Kantorovich averaging method. The radial and circumferential stresses due to

Chapter 2

the effects of large vibration amplitudes were presented with different values of the volume fraction index. Their research topic was further extended to cover thermal loading, force and moment on the vibration characteristics and stresses (Allahverdizadeh et al., 2007a, Allahverdizadeh et al., 2008) . The solution of large amplitude vibration of thick annular FG plates including thermal effect was also found in the literature (Amini et al., 2010, Tajeddini et al., 2011).

2.6 Functionally graded material fabrication

An extended review of techniques used for manufacturing FGMs had been presented by Mortensen and Suresh (Mortensen and Suresh, 1995). Several aspects were focused on in the review, for example underlying principles, critical issues in each process, mechanical analysis and performance of FGMs containing a metallic phase.

2.6.1 Thermal spraying technique

In the FGM fabrication of Kapuria et al.(Kapuria et al., 2008a) using thermal spraying technique, four stocks consisting of ceramic (Al2O3) volume fraction of 10%, 20%, 30% and 40% were prepared, and mechanical alloying was operated in a planetary mill. The rotation speed and the milling time were 150-250 rpm and 12 hrs, respectively. As a result, the expected size of the powder after alloying was around 20-50 μm which matched well with the introduction of Rabin and Heaps (Rabin and Heaps, 1992). To produce a 5 layered FGM sample as shown in Fig. 2.6, the Ni substrate surface was grit-blasted using SiC grits followed by ultrasonic cleaning in acetone. The stock materials in the form of powder were melted in an oxy-acetylene flame to create a fine spray. The dry acetylene pressure and oxygen pressure were kept as 1 kg f/ cm2 and 1.75 kg f/cm2 respectively. The materials were sprayed carefully on the whole surface of the Ni substrate. During the time of spraying the melted materials onto the substrate, the problem of substrate being damaged or distorted due to thermal stresses is possible. To overcome such problems, the chosen substrate would be thick enough to withstand a high temperature environment. Another difficulty which often happens when using the thermal spraying technique is the difficulty in controlling the uniformity of layered thickness. To understand clearly about the thermal

Chapter 2

spraying technique, the schematic of fabricating layered FGMs according to the technique is illustrated in Fig. 2.6.

Dry acetylene Dry acetylene and oxygen and oxygen

pressure pressure

Hot spray Hot spray

90%Ni-10% Al2O3 80%Ni-20% Al2O3

Ni substrate Ni substrate

Fig. 2.6 Schematic of thermal spraying technique for fabricating layered FGMs

According to the research of Kapuria et al. (Kapuria et al., 2008a), bending and free vibration analysis of layered functionally graded beams were considered in terms of theoretical and experimental investigations. Two types of the layered FG beams such as

Al/SiC beams and Ni/Al2O3 beams were produced using powder metallurgy and thermal spraying techniques, respectively. In the research, an introduction of the modified rule of mixtures (MROM) was presented in order to determine a more accurate prediction of effective modulus of elasticity. The beam geometries of the layered FG beams are shown in Fig. 2.7.

Chapter 2

Fig. 2.7 Geometry of multi-layered FGM beams (Kapuria et al., 2008a)

Kirihara et al. (Kirihara et al., 1997) fabricated Ti-Ti5Si3 FGMs by utilizing bonding through a eutectic reaction which occurs between Ti and Ti5Si3. The eutectic bonding method was used to bond a plate of pure Ti and Ti5Si3 together. The directed vapour deposition (DVD) technique was developed and explored as a potential FGM synthesis tool in order to enhance the quality of the spatially distributed microstructures of FGMs (Groves and Wadley, 1997). Porous Zirconia (ZrO2) preforms produced from a pyrolyzable pore forming agent (PFA) were used for the infiltration process with the molten metal to create functionally graded Al-Mg/ZrO2 components (Corbin et al., 1999). The technique using the pre-alloyed feedstock during plasma spraying was introduced in order to prepare FGMs with many advantages such as high deposition, more uniform coating density and improvement in chemical homogeneity (Gu et al., 1997, Khor and Gu, 2000). This technique was used in the study of Khor and Gu (Khor and Gu, 2000) to produce

ZrO2/NiCoCrAlY FGM coatings. The results using the effective technique can create the gradual changes of the coefficient of thermal expansion and thermal conductivity and diffusivity of the FGMs. In the study it was found that the thermal diffusivity and conductivity increase as the increment of NiCoCrAlY content and heating temperature.

Within the process of thermal spraying technique, the combustion of material powders in the technique can be achieved by using the flame spray process that was used to produce 54

Chapter 2

Ni/Al2O3 samples. To explain the main advantage of the process in comparison with the wire spray process, it is found that the flame spray process can be applied to a wider range of materials. However, it is suggested that the process is suitable for using with materials that have their melting temperature higher than the temperature of the flame.

According to the literature survey, the thermal spraying technique is one of the often used techniques, especially for the field of material coating technology and functionally graded material manufacturing. Another example is where ZrO2-NiCrAlY functionally graded material coatings were produced by using thermal-plasma spraying technique (Gu et al., 1997). Ceramic thermal barrier coating is the technology that can be used to improve the capacity of thermal resistance in materials. This technology is widely applied to make materials for gas turbine elements and engine parts. Two processes for producing thermal barrier coating were presented by Ma et al. (Ma et al., 2010). The electron beam physical vapour deposition (EB-PVD) and the atmospheric plasma spraying (APS) were demonstrated in the investigation. For instance, La2Ce2O7/8YSZ double-ceramic-layer thermal barrier coating system was produced by the atmospheric plasma spraying. The failure mechanism of the La2Ce2O7/8YSZ system due to the sintering of the La2Ce2O7 coating surface during thermal cycling was shown and discussed. Gorlach (Gorlach, 2009) developed a new method for making thermal spraying of Zn-Al coating. The new thermal spraying system was built on the principles of the high velocity air flame (HVAF) process. The development revealed that the significant advantage of HVAF sprayed coating was to enhance considerably higher bond strength than the coating produced by conventional methods. Optical microscopy, SEM, X-ray diffraction, bond strength and salt spray testing were adopted to examine the coated specimens. Consequently, it was found that coating with the new method led to a reduction in the number of porosities, and the quality of specimens has been improved in terms of corrosive resistance. In the study of Hou et al.

(Hou et al., 2011), NiAl-TiB2 composite coatings were produced by electro-thermal explosion ultrahigh speed spraying (EEUSS) technology. By using the EEUSS technology, the composite coatings have a very fine microstructure with submicron grains. Bonding between coating and substrate was represented as the metallic bond.

Chapter 2

2.6.2 Powder metallurgy technique

Powder metallurgy is also one of the useful techniques for fabricating functionally graded materials (FGMs). This is because the versatility inherent in the technique allows the making of the graded material compositions at the earliest stage in processing. To fabricate FGMs using the technique, the mixture of different material powders is stacked into a die as the desired sequence to obtain the material gradient, and then followed with pressing the mixed powders and sintering the compact. Kapuria et al. (Kapuria et al., 2008a) also used the powder metallurgy technique to produce layered FG beams. The beam samples were brought to test their behaviour of bending and vibration. The whole procedure of the powder metallurgy technique used for manufacturing the 5 layers-Al/SiC system is illustrated in Fig. 2.8.

Fundamentals of the graded structures or FGMs were discussed and emphasized in terms of basic principles of the materials and an approach of powder metallurgy techniques including hot isostatic processing used to make such materials (Yiasemides and Adkins, 1992). To investigate non-uniform sintering shrinkage of FGMs in the process of the powder metallurgy technique, Mizuno et al. (Mizuno et al., 1995) developed an “in situ monitoring system of sintering shrinkage” that uses digital image processing. By using the monitoring system, the outline of powder compacts during sintering was measured and captured in real time with acceptable results. FGM samples made of ZrO2 and NiCr were successfully manufactured by using the powder metallurgy technique in order to examine an effectiveness of the technique and the chemical composition as well as the microstructure across the material gradient. However, the shrinkage problem was found with the samples. The compacts with various powder mixing ratios led to different shrinkage (Zhu et al., 2001).

Chapter 2

(A) (B) (C) (D) (E)

100%A 90%Al 80%Al 70%Al 60%Al l0%SiC 10%SiC 20%SiC 30%SiC 40%SiC

Al-SiC powders in each bottle are mixed, dried, and blended for 12 hrs in ball milling machine with rotation speed of 150-250 rpm, and then lay up the powders into the die and punch as followings

E D C B A

The layered specimen is compressed in thickness direction with force of 490.4 kN to obtain the green compacted specimen

Sintering at 550 0C for 3.5 hrs

Fig. 2.8 Schematic of powder metallurgy technique to fabricate layered functionally graded beams made by Al/SiC

Chapter 2

In order to test thermal shock behaviour of FGMs, the powder metallurgy technique was used to produce mullite/Mo FGMs by Jin et al. (Jin et al., 2005) It was revealed that the FGM specimens had a greater capacity of thermal shock resistance than that of pure monolithic mullite specimens. In this study, several significant aspects, for example the FGM fabrication, microstructure and thermo-mechanical property analysis, thermal shock testing etc., were discussed in detail. For example, for the thermal shock testing, the FGM samples were heated by infrared radiation (IR) in which the temperature was controlled by adjusting the IR output power and the distance between the tip of the quartz rod and the samples. The heating time was around 20 s and then the samples were cooled suddenly by flowing cold water in order to observe cracks at the sample surfaces.

2.6.3 Infiltration technique

A multi-step sequential infiltration technique demonstrates its effectiveness in terms of producing FGMs composed of various pairs of materials. For example, using the infiltration technique, FGMs can be produced by a pair of ceramics-polymers or a pair of ceramics-metals. For a particular case of ceramic-polymer FGM, it can be achieved by using the infiltration technique, where this is impossible for the previous techniques. This is because the solidified temperature between polymer and ceramic is very different; hence, using the previous techniques, polymer phases are burnt out before the ceramic phases become solidified during the sintering process.

In the infiltration technique a sacrificial scaffold with open porosity such as a foam is infiltrated with the first phase. After this phase solidifies the scaffold is burnt away leaving a porous solid into which the second phase can be infiltrated.

Chapter 2

The earliest manufacture of graded composites using the infiltration technique was conducted by Cichocki et al. (Cichocki et al., 1998). In the investigations of Tilbrook et al. (Tilbrook et al., 2005, Tilbrook et al., 2006, Tilbrook et al., 2007), FGMs of alumina- epoxy, which have a large disparity in elastic properties, were produced by the technique for the purpose of investigating crack propagation. Methods used to predict effective mechanical properties of such FGMs were also included in those investigations. An example of graded specimen presented in the article of Tilbrook et al. (Tilbrook et al., 2006) is illustrated in Fig. 2.9 (a), which showed the two step-graded regions separated by the epoxy region and flanked by the ceramic-rich regions. To examine the quality of graded region of the specimen, examples of the composite microstructure were also illustrated in Fig. 2.9 (b). It can be seen that the quality of the specimen was excellent with well-bonded interfaces and a fewer number of pores were found inside the specimen. In some cases, Tilbrook et al. (Tilbrook et al., 2006) mentioned that delamination during the process of drying and burning out foam imprints led to imperfect interfaces between layers. In Fig. 2.9 (c), a series of layers were shown with several distinct layers, as observed from different shaded colours.

Chapter 2

(a) 95% Alumina 100% Epoxy 95% Alumina 5% Epoxy 5% Epoxy

(b) Layered graded areas (c)

30% epoxy 45% epoxy

70% alumina 55% alumina

Fig. 2.9 Layered graded composite specimen made of alumina and epoxy: (a) photograph of the graded specimen: (b) a single interface between layers: (c) a series of layers with several distinct layers (Tilbrook et al., 2006)

To make ceramic-metal FGMs using the infiltration technique, Neubrand et al. (Neubrand et al., 2002) produced functionally graded (FG) plates to deal with residual stress analysis. The finite element and experimental stress results were determined by sawing a notch in the plates and measuring displacements by Moire interferometry. The FG plate specimens made from Al/Al2O3 were manufactured by an adaptation of the gradient material by foam compaction (GMFC). The processing sequences of the GMFC were used to produce the FG plate specimens whose sizes were (35mm×33mm×3 mm). Details of the processing were presented in Fig. 2.10.

Chapter 2

¾ Polyurethane foam with

Open porosity

Cutting, Pressing

¾ Polyurethane foam with graded density

Casting of Al O slip into graded form, drying 2 3

¾ Al2O3/Polyurethane graded green body

Burnout foam, sintering

¾ Al 2O3 pre-form with graded porosity

Pressure assisted infiltration with Al melt

¾ Al/Al 2O3 FGM

Fig. 2.10 Processing sequences of infiltration technique using the GMFC

In this section, a detailed description on three types of FGM fabrication; i.e. thermal spraying, powder metallurgy and infiltration techniques has been made. A summary of their types, properties, production process and application is presented in Table 2.1. 61

Chapter 2

Table 2.1 General information of FGM types

Type of FGMs Properties Processing Applications Ceramic-Metal High toughness Thermal spraying Space vehicle Good thermal Powder metallurgy Engine combustion resistance Infiltration chamber Fusion reactor Ceramic-Polymer High flexibility Infiltration Electrical devices Good toughness Optics Good thermal and Biomedical electrical resistances Engineering

2.7 Conclusion

Based on the literature review as presented above, it is evident that many researchers have concentrated their attention on investigating behaviour of engineering structures made of FGMs for over the last decade. Due to several advantages of FGMs in terms of high strength and toughness as well as thermal and corrosive resistances, the materials have attracted interest from researchers continuously to further investigate more significant features. The power law distribution has been used widely to define effective material properties of FGMs. In order to predict more accurate behaviour of FG structures subjected to mechanical loads, higher shear deformation theories were selected to analyse bending, buckling and vibration problems for FG structures in past investigations. Within the development of beam and plate theories, the improved third order shear deformation theory (TSDT) proposed by Shi (Shi, 2007) was one of the effective theories, which was derived using more rigorous kinetics of displacements. The displacement and rotation functions of the theory were created using an elasticity formulation rather than the hypothesis of displacements. Therefore, the importance of using the improved TSDT over other HSDTs for the accurate prediction of beams and plates made from composite materials was presented and discussed by Shi (Shi, 2007). Very few reports dealing with thermal buckling analysis of FG beams and plates were found in the literature. Most of them presented only the results for temperature independent material properties.

Chapter 2

In this research, for the first time, the improved TSDT is used to analyse thermal buckling and elastic vibration of FG beams and plates. Two types of solutions that are the temperature independent solution and temperature dependent solution are employed to investigate both thermal buckling and elastic vibration of FG beams and plates under a high temperature environment. To validate numerical results of thermal buckling and vibration analysis presented in this research, a validation exercise is conducted and a comparison study is performed for the limited cases available in the open literature. Additionally, experiments are conducted for further validation.

As seen in the literature, there were only two reports of vibration of FG beams made of ceramic and metal that cover theoretical and experimental parts. Hence, in this research, FG beam specimens are fabricated using the multi-step sequential infiltration technique, to produce ceramic and polymer graded composite specimens for vibration testing. Several significant aspects such as boundary conditions, thickness and aspect ratio, material compositions, temperature dependency on material properties etc. which have impacts on thermal buckling and vibration results are investigated and discussed in detail. Finally, forced vibration analysis of FG plates subjected to uniformly dynamic loading is also included in this research.

Chapter 3 Development of analytical method using improved TSDT

Based on the extensive literature review for beam and plate theories, it is revealed that the third order shear deformation theories (TSDTs) account for a quadratic variation of the transverse shear stresses and strains across the thickness. TSDTs usually provide more accurate stress distributions without any requirement of shear correction factors. By ,expanding the displacements ሺݑǡݒǡݓሻ as cubic functions of the thickness coordinate ሺݖሻ the straightness and normality of a transverse normal after deformation are relaxed. To investigate behaviour of engineering structures made of FGMs, it is expected that TSDTs can yield more accurate results. This is due to the nonlinear material profiles of FGM structures across their thickness matching well with the cubic displacement functions used in TSDTs. Hence, in this research, TSDTs are applied to investigate thermal buckling and elastic vibration response of beams and plates made by FGMs.

An improved third order shear deformation theory was developed by Shi (Shi, 2007) based on rigorous kinematics of displacements. The theory was used to analyse static analysis of isotropic and orthotropic beams and plates. The author concluded that the new theory provided more accurate results than those obtained from other higher order shear deformable theories, especially when the transverse shear plays a very important role. Because of the kinematics of displacement in the new TSDT derived from an elasticity formulation rather than the hypothesis of displacements, it is interesting to further implement this theory for the thermal buckling and free vibration analysis of the heated 64

Chapter 3

functionally graded (FG) beams and plates in this research. The coordinate systems for beam and plate that will be used throughout this research are defined in Fig. 3.1.

z Plate Beam z y y x

Fig. 3.1 The coordinate systems of beam and plate

To apply the improved theory for analysing beam and plate problems, it is begun with the constitutive equations that take the form as,

ߪ௫௫ ܳଵଵ ܳଵଶ Ͳ ߳௫௫ ߬௬௭ ܳସସ Ͳ ߛ௬௭ ൝ߪ௬௬ൡൌ൥ܳଶଵ ܳଶଶ Ͳ ൩൝߳௬௬ൡǡቄ ቅൌ൤ ൨ቄ ቅǤ ߬௫௭ Ͳܳହହ ߛ௫௭ (3.1) ߬௫௬ ͲͲܳ଺଺ ߛ௫௬ where ܳ௜௝ are the plane stress reduced elastic constants in material axes of the plate.

For general orthotropic plate, the components of ܳ௜௝are defined as,

ܧଵ ߥଵଶܧଶ (3.2a) ܳଵଵ ൌ ǡܳଵଶ ൌ ͳെߥଵଶߥଶଵ ͳെߥଵଶߥଶଵ

ܧଶ (3.2b) ܳଶଶ ൌ ǡܳ଺଺ ൌܩଵଶǡܳସସ ൌܩଶଷǡܳହହ ൌܩଵଷ ͳെߥଵଶߥଶଵ

For FG plate, the components of ܳ௜௝are defined as,

(ሺݖǡ ܶሻ (3.3aܧሺݖǡ ܶሻ ߥܧ ܳ ൌܳ ൌ ǡܳ ൌܳ ൌ ǡ ଵଵ ଶଶ ͳെߥଶ ଵଶ ଶଵ ͳെߥଶ

(ሺݖǡ ܶሻ (3.3bܧ ሺݖሻ ൌ Ǥܩൌܳ ൌܳ ൌ ܳ ସସ ହହ ଺଺ ʹሼͳ൅ߥሽ The displacement field of the improved TSDT is written as follows,

Chapter 3

(߲ݓ଴ ܳ௫ Ͷ ଶ (3.4a ݑൌݑ଴ሺݔǡ ݕǡ ݐሻ െ ݖ൅ ൬͵ݖ െ ଶ ݖ ൰ǡ ݄ ߲ݔ ʹܳହହ݄

ܳ (߲ݓ଴ ௬ Ͷ ଶ (3.4b ݒൌݑ଴ሺݔǡ ݕǡ ݐሻ െ ݖ൅ ൬͵ݖ െ ଶ ݖ ൰ǡ ݄ ߲ݕ ʹܳସସ݄

(ݓൌݓ଴ሺݔǡ ݕǡ ݐሻǤ (3.4c

ହ డ௪ ହ డ௪ where ܳ ൌ ܳ ݄ቀ߶ ൅ బቁ and ܳ ൌ ܳ ݄ቀ߶ ൅ బቁ. ௫ ଺ ହହ ௫ డ௫ ௬ ଺ ସସ ௬ డ௬

It is noted that ܳ௫ and ܳ௬ are the shear forces acting on the cross-sections with the normal డ௪ in the x and y coordinates; moreover, the physical meanings of ቀ߶ ൅ బቁ and ቀ߶ ൅ ௫ డ௫ ௬ డ௪ బቁ terms are the transverse shears of the cross-section with x=constant and y=constant, డ௬ ହ respectively. The factor is obtained from the work equivalence between the transverse ଺ shear forces and the transverse shear stresses. Therefore, the displacement field used for the following calculation can then be expressed as;

(ͷ Ͷ ͳ ͷ ߲ݓ (3.5a ݑൌݑ ሺݔǡ ݕǡ ݐሻ ൅ ൬ݖ െ ݖଷ൰߶ ሺݔǡ ݕǡ ݐሻ ൅൬ ݖെ ݖଷ൰ ଴ ଴ Ͷ ͵݄ଶ ௫ Ͷ ͵݄ଶ ߲ݔ

(ͷ Ͷ ଷ ͳ ͷ ଷ ߲ݓ଴ (3.5b ݒൌݒ଴ሺݔǡ ݕǡ ݐሻ ൅ ൬ݖ െ ଶ ݖ ൰߶௬ሺݔǡ ݕǡ ݐሻ ൅൬ ݖെ ଶ ݖ ൰ Ͷ ͵݄ Ͷ ͵݄ ߲ݕ

(ݓൌݓ଴ሺݔǡ ݕǡ ݐሻǤ (3.5c

In which ݑ଴ǡݒ଴ǡݓ଴ǡ߶௫ and ߶௬ are generalised displacements at the reference surface of plateሺݖ ൌ Ͳሻ. ߶௫ and ߶௬ are slopes of the transverse normal about the y and x axes at ݖൌ Ͳ, and t is the time. Due to the advantages of the improved TSDT as presented above, the theory is employed to analyse mechanical problems of functionally graded structures in this chapter. The development of analytical methods using the improved TSDT is presented for the following in this chapter:

Chapter 3

(a) Thermal buckling and elastic vibration analysis of FG beams

(b) Thermal buckling and elastic vibration analysis of FG plates

3.1 (a) Thermal buckling and elastic vibration analysis of FG beams

To consider functionally graded (FG) beam analysis, stress and strain components in relation to the y coordinate can be neglected, so the displacement field of the improved TSDT used for beam analysis can be written as follows:

(ͷ Ͷ ͳ ͷ ߲ݓ (3.6a ݑൌݑ ሺݔǡ ݕǡ ݐሻ ൅ ൬ݖ െ ݖଷ൰߶ ሺݔǡ ݕǡ ݐሻ ൅൬ ݖെ ݖଷ൰ ଴ ଴ Ͷ ͵݄ଶ ௫ Ͷ ͵݄ଶ ߲ݔ

(ݓൌݓ଴ሺݔǡ ݕǡ ݐሻǤ (3.6b

From the displacement field in Eq. (3.6), the small normal strainሺߝ௫௫) and the transverse shear strainሺߛ௫௭ሻ can be expressed as:

(߲ݑ (3.7 ߝ ൌ ൌߝሺ଴ሻ ൅ݖߝሺଵሻ ൅ݖଷߝሺଷሻǡ ௫௫ ߲ݔ ௫௫ ௫௫ ௫௫

(߲ݑ ߲ݓ (3.8 ߛ ൌ ൅ ൌߛሺ଴ሻ ൅ݖଶߛሺଶሻǤ ௫௭ ߲ݖ ߲ݔ ௫௭ ௫௭

The strain components of TSDT can be obtained by substituting the displacement field from Eq. (3.6) into the strain equations in Eqs. (3.7-3.8). Therefore, the normal and shear strain components can be expressed as:

డ௨ ߝሺ଴ሻ ൌ బ ௫௫ డ௫ ۓ మ ۖ ሺଵሻ ଵ డథೣ డ ௪బ (3.9) Normal strain components ߝ௫௫ ൌ ቀͷ ൅ ቁ ସ డ௫ డమ௫ ۔ ିହ డథ డమ௪ ߝሺଷሻ ൌ ቀ ೣ ൅ బቁǡۖ ௫௫ ଷ௛మ డ௫ డమ௫ ە

Chapter 3

ହ డ௪ ߛሺ଴ሻ ൌ ቀ߶ ൅ బቁ ௫௭ ସ ௫ డ௫ Shear strain components ቐ ିହ డ௪ (3.10) ߛሺଶሻ ൌ ቀ߶ ൅ బቁǤ ௫௭ ௛మ ௫ డ௫

The relationship between stresses and strains for TSDT in the form of elastic constitutive equations is:

ሺ଴ሻ ሺଵሻ ଷ ሺଷሻ (3.11a) ߪ௫௫ ൌܳଵଵሺݖሻቂߝ௫௫ ൅ݖߝ௫௫ ൅ݖ ߝ௫௫ ቃǡ

ሺ଴ሻ ଶ ሺଶሻ (3.11b) ߬௫௭ ൌܳହହሺݖሻቂߛ௫௭ ൅ݖ ߛ௫௭ ቃǤ

The elastic constants for the layered FG beams varied through the beam thickness can be expressed as:

(ሺݖሻ (3.12ܧ ሺݖሻܧ ሺݖሻ ൌ Ǥܩሺݖሻ ൌ ǡܳ ൌ ܳ ଵଵ ͳെߥଶ ହହ ʹሺͳ൅ߥሻ

3.1.1 Strain energy for FG beams

The total strain energy of FG beams due to the normal force, shear force, moment and higher order terms can be expressed as:

௅ ͳ ߝሺଵሻ ൅ܲ ߝሺଷሻቃ൅ቂܳ ߛሺ଴ሻ ൅ܴ ߛሺଶሻቃ൨ ݀ݔǡ ܯൌ න൤ቂܰ ߝሺ଴ሻ ൅ ܷ ௘ ʹ ௫௫ ௫௫ ௫௫ ௫௫ ௫௫ ௫௫ ௫ ௫௭ ௫ ௫௭ (3.13) ଴

where

௛ ௛ ଶ ଶ ܰ௫௫ ͳ ܳ௫ ͳ (3.14) ௫௫ൡൌ න൝ݖ ൡ ߪ௫௫݀ݖǡ ൜ ൠൌ නቄ ଶቅ ߬௫௭݀ݖǤܯ൝ ଷ ܴ௫ ݖ ௫௫ ି௛ ݖ ି௛ܲ ଶ ଶ

It is noted that the terms that multiply with the normal and shear strain components can be defined as follows:

Chapter 3

ܰ௫௫ is the normal force resultant.

ܯ௫௫ is the moment resultant.

ܲ௫௫ is the higher-order stress resultant.

ܳ௫ is the shear force resultant.

ܴ௫ is the higher-order shear stress resultant.

Substituting Eq. (3.14) into Eq. (3.13), the strain energy expression can be obtained as a function of material stiffness and strain components:

௅ ͳ ʹ ʹ ܷ ൌ න ቂܣ ߝሺͲሻ ൅ʹܤ ߝሺͲሻߝሺͳሻ ൅ܦ ߝሺͳሻ ൅ʹܧ ߝሺͲሻߝሺ͵ሻ ൅ʹܨ ߝሺͳሻߝሺ͵ሻ (௘ ʹ ͳͳ ݔݔ ͳͳ ݔݔ ݔݔ ͳͳ ݔݔ ͳͳ ݔݔ ݔݔ ͳͳ ݔݔ ݔݔ (3.15 ଴

ሺ͵ሻʹ ሺͲሻʹ ሺͲሻ ሺʹሻ ሺʹሻʹ ͷͷߛݔݖ ቃ ݀ݔǡܨͷͷߛݔݖ ߛݔݖ ൅ܦʹͷͷߛݔݖ ൅ܣͳͳߝݔݔ ൅ܪ൅

where A11, B11, D11, E11, F11, H11, A55, D55 and F55 are the material stiffness components. In order to define a strain energy equation that is associated with TSDT in the form of displacement and rotation functions, Eqs. (3.9-3.10) can be substituted into Eq. (3.15). Therefore, the strain energy equation for the TSDT becomes:

௅ ʹ ʹ ʹ ʹ ͳͳ ߲߶ݔ ߲ ݓͲܦ ͳͳ ߲ݑͲ ߲߶ݔ ߲ݑͲ ߲ ݓͲܤ ͳ ߲ݑ݋ ܷ௘ ൌ න ቎ܣͳͳ ቆ ቇ ൅ ൭ͷ ൅ ൱ ൅ ൭ͷ ൅ ൱ ߲ݔ ʹ ߲ݔ ߲ݔ ߲ݔ ߲ʹݔ ͳ͸ ߲ݔ ߲ʹݔ ʹ ଴ ʹ ͳͳ ߲ݑͲ ߲߶ݔ ߲ݑͲ ߲ ݓͲܧͳͲ െ ൭ ൅ ൱ ߲ݔ ߲ݔ ߲ݔ ߲ʹݔ ʹ݄͵

ʹ ʹ ʹ ʹ ͳͳ ͷ ߲߶ ߲߶ ߲ ݓͲ ͳ ߲ ݓͲܨͳͲ െ ቌ ቆ ݔቇ ൅ ݔ ൅ ൭ ൱ ቍ ͸ ߲ݔ ߲ݔ ߲ʹݔ ͸ ߲ʹݔ ʹ݄ʹ (3.16) ʹ ʹ ʹ ͷͷ ߲ݓͲܣͳͳ ߲߶ ߲ ݓͲ ʹͷܪͷʹ ൅ ൭ ݔ ൅ ൱ ൅ ቆ߶ ൅ ቇ ͻ݄Ͷ ߲ݔ ߲ʹݔ ͳ͸ ݔ ߲ݔ

ʹ ʹ ͷͷ ߲ݓͲܨͷͷ ߲ݓͲ ʹͷܦͷʹ െ ቆ߶ ൅ ቇ ൅ ቆ߶ ൅ ቇ ቏ ݀ݔǤ ݔ ߲ݔ ݄Ͷ ݔ ߲ݔ ʹ݄ʹ

Chapter 3

The material stiffness components used in Eq. (3.16) can be defined as the extensional stiffness A11, bending-coupling stiffness B11, bending stiffness D11, warping-extensional coupling stiffness E11, warping-bending coupling stiffness F11, and warping-higher order bending coupling stiffness H11. A55, D55 and F55 are associated with shear stiffness components. The stiffness components can be expressed as:

݄ ʹ ଶ ଷ ସ ଺ (3.17a) ଵଵሻ ൌ නܳଵଵሺݖሻሺͳǡ ݖǡ ݖ ǡݖ ǡݖ ǡݖ ሻ݀ݖǡܪଵଵǡܨଵଵǡܧଵଵǡܦଵଵǡܤଵଵǡܣሺ െ݄ ʹ

݄ ʹ ଶ ସ (3.17b) ହହሻ ൌ නܳହହሺݖሻሺͳǡ ݖ ǡݖ ሻ݀ݖǤܨହହǡܦହହǡܣሺ െ݄ ʹ

The effective material properties of FGMs based on the power law distribution can be obtained from the following equations,

(ݖ ͳ ௡ (3.18a ǡ ܧሻ ൬ ൅ ൰ ൅ ܧെ ܧሺݖሻ ൌ ሺܧ ௧ ௕ ݄ ʹ ௕

(ݖ ͳ ௡ (3.18b ߙሺݖሻ ൌ ሺߙ െߙ ሻ ൬ ൅ ൰ ൅ߙ ǡ ௧ ௕ ݄ ʹ ௕

(ݖ ͳ ௡ (3.18c ߩሺݖሻ ൌ ሺߩ െߩ ሻ ൬ ൅ ൰ ൅ߩ ǡ ௧ ௕ ݄ ʹ ௕

The subscripts t and b mean the material properties at the top and bottom beam surfaces, ሺݖሻ is Young’s modulus, ߙሺݖሻ the coefficient of thermalܧ respectively. It is also defined that .expansion, ߩሺݖሻ material density and the constant ߥ is Poisson’s ratio

By using Eq. (3.17) with the equations of the effective material properties of FGMs as presented in Eq. 3.18, one can obtain the material stiffness components for normal and shear stresses in the form of the material volume fraction index (n), using integration by parts and substitution. This form of material stiffness components is more convenient for subsequent investigation. The expanded terms are shown as follows:

Chapter 3

݄ ܧ௧௕ (3.19a) ܣ ൌ ൤ ൅ܧ ൨ ଵଵ ͳെߥଶ ሺ݊൅ͳሻ ௕

ܧ ݄ଶ ݊ (3.19b) ܤ ൌ ௧௕ ൤ ൨ ଵଵ ͳെߥଶ ʹሺ݊൅ͳሻሺ݊൅ʹሻ

݄ଷ ܧ ሺ݊ଶ ൅݊൅ʹሻ ܧ (3.19c) ܦ ൌ ቈ ௧௕ ൅ ௕ ቉ ଵଵ ͳെߥଶ Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻ ͳʹ

ܧ ݄ସ ݊ሺ݊ଶ ൅͵݊൅ͺሻ (3.19d) ܧ ൌ ௧௕ ቈ ቉ ଵଵ ͳെߥଶ ͺሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻ

݄ହ ܧ ሺ݊ସ ൅͸݊ଷ ൅ ʹ͵݊ଶ ൅ ͳͺ݊ ൅ ʹͶሻ ܧ (3.19e) ܨ ൌ ቈ ௧௕ ൅ ௕቉ ଵଵ ͳെߥଶ ͳ͸ሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻ ͺͲ

݄଻ ܧ ሺ݊଺ ൅ͳͷ݊ହ ൅ ͳͳͷ݊ସ ൅ ͶͲͷ݊ଷ ൅ ͻ͸Ͷ݊ଶ ൅ ͸͸Ͳ݊ ൅ ͹ʹͲሻ (3.19f) ܪ ൌ ቈ ௧௕ ଵଵ ͳെߥଶ ͸Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻሺ݊൅͸ሻሺ݊൅͹ሻ ܧ ൅ ௕ ቉ ͶͶͺ

݄ ܧ௧௕ (3.20a) ܣ ൌ ൤ ൅ܧ ൨ ହହ ʹሺͳ ൅ ߥሻ ሺ݊൅ͳሻ ௕

݄ଷ ܧ ሺ݊ଶ ൅݊൅ʹሻ ܧ (3.20b) ܦ ൌ ቈ ௧௕ ൅ ௕ ቉ ହହ ʹሺͳ ൅ ߥሻ Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻ ͳʹ

݄ହ ܧ ሺ݊ସ ൅͸݊ଷ ൅ʹ͵݊ଶ ൅ ͳͺ݊ ൅ ʹͶሻ ܧ (3.20c) ܨ ൌ ቈ ௧௕ ൅ ௕቉ ହହ ʹሺͳ ൅ ߥሻ ͳ͸ሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻ ͺͲ

where ܧ௧௕ ൌܧ௧ െܧ௕Ǥ The detailed definition of parameters in the equations of material stiffness can be seen in Chapter 4.

Chapter 3

3.1.2 The potential energy for FG beams due to thermal stresses

For a typical FG beam which has been in a high temperature environment for a long period of time, it is assumed that the temperature can be distributed uniformly across its thickness so that the case of uniform temperature rise is taken into consideration. In this investigation,

initial uniform temperature is ܶ଴ (ܶ଴ ൌ ͵ͲͲܭሻ , which is a stress free state, changes to final temperature with ȟܶ . For inclusion in the improved TSDT the thermal stress for a constrained beam with no deflection is required. This stress can be defined as

(ሺݖሻߙሺݖሻȟܶ (3.21ܧ ߪ் ൌߪ் ൌെ ǡ்߬ ൌ்߬ ൌͲ ௫௫ ͳെߥ ௫௬

The potential energy due to thermal stress can be written as (Kim, 2005)

௅ ʹ ʹ ܶ ʹ ͳͳ ߲ݑͲ ߲߶ݔ ߲ݑͲ ߲ ݓͲܤ ͳ ܶ ߲ݑͲ ߲ݓͲ ܸ௘ ൌ න ቎ܣͳͳ ൭ቆ ቇ ൅ ቆ ቇ ൱ ൅ ൭ͷ ൅ ൱ ߲ݔ ߲ݔ ʹ ߲ݔ ߲ݔ ߲ݔ ߲ʹݔ ʹ ଴

ܶ ʹ ʹ ʹ ʹ ͳͳ ߲߶ ߲߶ ߲ ݓͲ ߲ ݓͲܦ ൅ ቌʹͷ ቆ ݔቇ ൅ͳͲ ݔ ൅ ൭ ൱ ቍ ͳ͸ ߲ݔ ߲ݔ ߲ʹݔ ߲ʹݔ

߲ݑ ߲߶ ߲ݑ ߲ʹݓ ܶܧͳͲ (െ ͳͳ ൭ Ͳ ݔ ൅ Ͳ Ͳ൱ (3.22 ߲ݔ ߲ݔ ߲ݔ ߲ʹݔ ʹ݄͵

ܶ ʹ ʹ ʹ ʹ ͳͳ ߲߶ ߲߶ ߲ ݓͲ ߲ ݓͲܨͳͲ െ ቌͷ ቆ ݔቇ ൅͸ ݔ ൅ ൭ ൱ ቍ ͳʹ݄ʹ ߲ݔ ߲ݔ ߲ʹݔ ߲ʹݔ

ܶ ʹ ʹ ʹ ʹ ͳͳ ߲߶ ߲߶ ߲ ݓͲ ߲ ݓͲܪͷʹ ൅ ቌቆ ݔቇ ൅ʹ ݔ ൅ ൭ ൱ ቍ቏ ݀ݔǤ ͻ݄Ͷ ߲ݔ ߲ݔ ߲ʹݔ ߲ʹݔ

Where the thermal stress and the higher order terms can be written as,

௛ ଶ ሻ ൌන ߪ்ሺͳǡ ݖǡ ݖଶǡݖଷǡݖସǡݖ଺ሻ ݀ݖǤ ்ܪǡ ்ܨǡ ்ܧǡ ்ܦǡ ்ܤǡ ்ܣሺ ଵଵ ଵଵ ଵଵ ଵଵ ଵଵ ଵଵ ି௛ (3.23) ଶ

Again for material stiffness due to thermal stresses, the same integration techniques are applied to obtain the material stiffness components that can be written in the form of parameter n as:

Chapter 3

݄οܶ ܧ௧௕ߙ௧௕ ܧ௧௕ߙ௕ ൅ܧ௕ߙ௧௕ (3.24a) ܣ் ൌ ൤ ൅ ൅ܧ ߙ ൨ ଵଵ ͳെߥ ሺʹ݊ ൅ ͳሻ ሺ݊ ൅ ͳሻ ௕ ௕

݄݊ଶοܶ ܧ ߙ ܧ ߙ ൅ܧ ߙ (3.24b) ܤ் ൌ ൤ ௧௕ ௧௕ ൅ ௧௕ ௕ ௕ ௧௕ ൨ ଵଵ ͳെߥ ʹሺ݊ ൅ ͳሻሺʹ݊ ൅ ͳሻ ʹሺ݊൅ͳሻሺ݊ ൅ ʹሻ

݄ଷοܶ ܧ ߙ ሺʹ݊ଶ ൅݊൅ͳሻ ሺܧ ߙ ൅ܧ ߙ ሻሺ݊ଶ ൅݊൅ʹሻ (3.24c) ܦ் ൌ ቈ ௧௕ ௧௕ ൅ ௧௕ ௕ ௕ ௧௕ ଵଵ ͳെߥ ʹሺʹ݊ ൅ ͳሻሺʹ݊ ൅ ʹሻሺʹ݊ ൅ ͵ሻ Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊ ൅ ͵ሻ ܧ ߙ ൅ ௕ ௕቉ ͳʹ

ସ ଶ ் ݄݊ οܶ ܧ௧௕ߙ௧௕ሺʹ݊ ൅͵݊൅Ͷሻ ܧଵଵ ൌ ቈ ͳെߥ ʹሺʹ݊ ൅ ͳሻሺʹ݊ ൅ ʹሻሺʹ݊ ൅ ͵ሻሺʹ݊ ൅ Ͷሻ (3.24d) ሺܧ ߙ ൅ܧ ߙ ሻሺ݊ଶ ൅͵݊൅ͺሻ ൅ ௧௕ ௕ ௕ ௧௕ ቉ ͺሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊ ൅ Ͷሻ

ହ ସ ଷ ଶ ் ݄ οܶ ܧ௧௕ߙ௧௕ሺͶ݊ ൅ͳʹ݊ ൅ʹ͵݊ ൅ͻ݊൅͸ሻ ܨଵଵ ൌ ቈ ͳെߥ Ͷሺʹ݊ ൅ ͳሻሺʹ݊ ൅ ʹሻሺʹ݊ ൅ ͵ሻሺʹ݊ ൅ Ͷሻሺʹ݊ ൅ ͷሻ (3.24e) ሺܧ ߙ ൅ܧ ߙ ሻሺ݊ସ ൅͸݊ଷ ൅ ʹ͵݊ଶ ൅ ͳͺ݊ ൅ ʹͶሻ ܧ ߙ ൅ ௧௕ ௕ ௕ ௧௕ ൅ ௕ ௕቉ ͳ͸ሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊ ൅ ͷሻ ͺͲ

் ܪଵଵ ݄଻οܶ ܧ ߙ ሺͺ݊଺ ൅͸Ͳ݊ହ ൅ ʹ͵Ͳ݊ସ ൅ ͶͲͷ݊ଷ ൅ Ͷͺʹ݊ଶ ൅ ͳ͸ͷ݊ ൅ ͻͲሻ ൌ ቈ ௧௕ ௧௕ ͳെߥ ͺሺʹ݊ ൅ ͳሻሺʹ݊ ൅ ʹሻሺʹ݊ ൅ ͵ሻሺʹ݊ ൅ Ͷሻሺʹ݊ ൅ ͷሻሺʹ݊ ൅ ͸ሻሺʹ݊ ൅ ͹ሻ (3.24f) ሺܧ ߙ ൅ܧ ߙ ሻሺ݊଺ ൅ͳͷ݊ହ ൅ ͳͳͷ݊ସ ൅ ͶͲͷ݊ଷ ൅ ͻ͸Ͷ݊ଶ ൅ ͸͸Ͳ݊ ൅ ͹ʹͲሻ ൅ ௧௕ ௕ ௕ ௧௕ ͸Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻሺ݊൅͸ሻሺ݊ ൅ ͹ሻ ܧ ߙ ൅ ௕ ௕቉ ͶͶͺ

where ߙ௧௕ ൌߙ௧ െߙ௕.

Chapter 3

3.1.3 The kinetic energy for FG beams

To analyse thermo-elastic vibration, the kinetic energy of FG beam is required which can then be expressed as,

௛ ௅ ଶ ʹ ʹ ͳ ߲ݑ ߲ݓ (ൌ නනߩሺݖሻ ൥ቆ ቇ ൅ ቆ ቇ ൩ ݀ݖ݀ݔǡ (3.25 ܶ ௘ ʹ ߲ݐ ߲ݐ ଴ ି௛ ଶ .where ߩሺݖሻ is the mass density per unit volume that varies across the thickness direction The displacement functions in Eq. (3.6) are substituted into Eq. (3.25) to produce the kinetic energy equations for TSDT as following,

௅ ʹ ʹ ʹ ͳ ߲ݑͲ ߲ݓͲ ͳ ߲ݑͲ ߲߶ ߲ݑͲ ߲ ݓͲ ܫ ൅ ൭ͷ ݔ ൅ ൱ ܫ ൌ න ቎൭ቆ ቇ ൅ ቆ ቇ ൱ ܶ ௘ ʹ ߲ݐ ߲ݐ Ͳ ʹ ߲ݐ ߲ݐ ߲ݐ ߲ݔ߲ݐ ͳ ଴ ʹ ʹ ʹ ʹ ͳ ߲߶ ߲߶ ߲ ݓͲ ߲ ݓͲ ܫ ൅ ቌʹͷ ቆ ݔቇ ൅ͳͲ ݔ ൅ ൭ ൱ ቍ ʹ ͳ͸ ߲ݐ ߲ݐ ߲ݔ߲ݐ ߲ݔ߲ݐ ʹ ͳͲ ߲ݑͲ ߲߶ݔ ߲ݑͲ ߲ ݓͲ െ ൭ ൅ ൱ ܫ͵ (߲ݐ ߲ݐ ߲ݐ ߲ݔ߲ݐ (3.26 ʹ݄͵ ʹ ʹ ʹ ʹ ͳͲ ߲߶ݔ ߲߶ݔ ߲ ݓͲ ߲ ݓͲ െ ቌͷ ቆ ቇ ൅͸ ൅ ൭ ൱ ቍ ܫͶ ͳʹ݄ʹ ߲ݐ ߲ݐ ߲ݔ߲ݐ ߲ݔ߲ݐ

ʹ ʹ ʹ ʹ ͷ ߲߶ݔ ߲߶ݔ ߲ ݓͲ ߲ ݓͲʹ ͸቏ ݀ݔǡܫ ൅ ቌቆ ቇ ൅ʹ ൅ ൭ ൱ ቍ ͻ݄Ͷ ߲ݐ ߲ݐ ߲ݔ߲ݐ ߲ݔ߲ݐ

where

௛ ଶ ݅ (௜ ൌනߩሺݖሻݖ ݀ݖ Ǣ݅ ൌ Ͳǡͳǡʹǡ͵ǡͶǡ͸Ǥ (3.27ܫ ି௛ ଶ

Chapter 3

The components of moment of inertia (ܫ௜) can be presented in function of value n as:

ߩ௧௕ (3.28a) ܫ ൌ݄൤ ൅ߩ ൨ ଴ ሺ݊൅ͳሻ ௕

݊ (3.28b) ܫ ൌ݄ଶߩ ൤ ൨ ଵ ௧௕ ʹሺ݊൅ͳሻሺ݊൅ʹሻ

ߩ ሺ݊ଶ ൅݊൅ʹሻ ߩ (3.28c) ܫ ൌ݄ଷ ቈ ௧௕ ൅ ௕ ቉ ଶ Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻ ͳʹ

݊ሺ݊ଶ ൅͵݊൅ͺሻ (3.28d) ܫ ൌ݄ସߩ ቈ ቉ ଷ ௧௕ ͺሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻ

ߩ ሺ݊ସ ൅͸݊ଷ ൅ʹ͵݊ଶ ൅ ͳͺ݊ ൅ ʹͶሻ ߩ (3.28e) ܫ ൌ݄ହ ቈ ௧௕ ൅ ௕ ቉ ସ ͳ͸ሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻ ͺͲ

ߩ ሺ݊଺ ൅ ͳͷ݊ହ ൅ ͳͳͷ݊ସ ൅ ͶͲͷ݊ଷ ൅ ͻ͸Ͷ݊ଶ ൅ ͸͸Ͳ݊ ൅ ͹ʹͲሻ ߩ (3.28f) ܫ ൌ݄଻ ቈ ௧௕ ൅ ௕ ቉ ଺ ͸Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻሺ݊൅͸ሻሺ݊൅͹ሻ ͶͶͺ where ߩ௧௕ ൌߩ௧ െߩ௕.

The total energy functional ሺȫሻ of FG beams for the thermal buckling analysis can then be written in the following form:

ȫൌܷ௘ ൅ܸ௘Ǥ (3.29)

For the case of vibration analysis including thermal effects, the total energy functional ሺȫሻ of FG beams is

ȫൌܷ௘ ൅ܸ௘ െܶ௘Ǥ (3.30)

Chapter 3

3.1.4 The solution method for FG beam analysis

A variational approach, the Ritz method, is employed to determine the thermal buckling and natural frequency parameters of the FG beams based on the improved TSDT. The FG beams with movable and immovable boundary conditions, for example, simply-free (S-F), hinged-hinged (H-H), clamped-clamped (C-C), etc. are chosen to consider in this research. This method requires the admissible functions for displacement and rotation components

.ݑ଴ሺݔǡ ݐሻ, ݓ଴ሺݔǡ ݐሻ and ߶௫ሺݔǡ ݐሻ that must satisfy at least the essential boundary conditions Otherwise, inappropriate functions lead to very slow convergence rates and unstable results. The Ritz method is known as an effective tool that has been employed to solve vibration problems by many researchers (Kitipornchai et al., 2009, Aydogdu, 2005, Chen et al., 1999, Huang and Leissa, 2009). In this investigation, the algebraic polynomial trial functions are selected to find out the thermal buckling and natural frequency results of the FG beams. It is very important to use a sufficient number of terms with the Ritz method in order to obtain an accurate solution (Huang and Leissa, 2009, Qatu, 2009). Therefore, the polynomial trial functions for the beam analysis are expressed in the following forms,

ॶ (3.31a) ঋబ ঐబ ݑ଴ሺߦǡ ݐሻ ൌ෍ܷঋߦ ሺߦെͳሻ ǡ

ঋୀঋబ

ॷ (3.31b) ঌబ ঑బ ߶௫ሺߦǡ ݐሻ ൌ෍Ȱঌߦ ሺߦെͳሻ ǡ

ঌୀঌబ

ॸ (3.31c) ঍బ ঒బ ݓ଴ሺߦǡ ݐሻ ൌ෍ܹ঍ߦ ሺߦെͳሻ Ǥ

঍ୀ঍బ

௫ Where ߦൌ is a non-dimensional coordinate and ܷ ǡȰ and ܹ are the unknown ௅ ঋ ঌ ঍ coefficients. In this research, the upper limit number of the polynomial trial functions ॶ, ॷ and ॸ should be set to be the same number as Գ. The displacement and rotation functions in Eq. (3.31) for each type of boundary condition has to satisfy the kinematic boundary conditions presented below, 76

Chapter 3

(Simply supported (S): ݑ଴ ്ͲǢ߶௫ ്ͲǢݓ଴ ൌͲ (3.32a

(Hinged (H): ݑ଴ ൌͲǢ߶௫ ്ͲǢݓ଴ ൌͲ (3.32b ௗ௪ (Clamped (C): ݑ ൌ߶ ൌݓ ൌ బ ൌͲ (3.32c ଴ ௫ ଴ ௗక

(Free (F): ݑ଴ ്ͲǢ߶௫ ്ͲǢݓ଴ ്Ͳ (no constraint conditions) (3.32d

Several configurations of boundary conditions considered in this research for FG beam analysis are illustrated in Fig. 3.2.

Simply supported (S)

Hinged (H)

Clamped (C)

Free (F)

Fig. 3.2 End conditions of FG beams

It can be noted that the hinged (H) boundary condition, in which its in-plane displacement is constrained, is a special case of simply supported (S) boundary condition. The H boundary condition is by default on the geometric mid-plane of the beam. To achieve computation for beam analysis with various boundary conditions using the Ritz method, Table 3.1 gives the displacement and rotation field indices of the method which are used for Eq. (3.31).

Chapter 3

Table 3.1 Displacement and rotation field indices for the Ritz method

At ߦൌͲ At ߦൌͳ

B.C. ঋ଴ ঌ଴ ঍଴ ঐ଴ ঑଴ ঒଴ C-F 1 1 2 0 0 0 S-S 0 0 1 0 0 1 C-S 1 1 2 0 0 1 C-C 1 1 2 1 1 2 S-F 0 0 1 0 0 0 F-F 0 0 0 0 0 0 H-H 1 0 1 1 0 1 H-C 1 0 1 1 1 2

Substituting the polynomial trial functions of Eq. (3.31) into the total energy functional ሺȫሻ in Eq. (3.29) for thermal buckling analysis and then taking derivative with respect to the unknown coefficients in the procedure of minimisation requires

߲ȫ ߲ȫ ߲ȫ (3.33) ൌͲǡ ൌͲǡ ൌͲǤ ߲ܷঋ ߲Ȱঌ ߲ܹ঍

This procedure leads to a system of simultaneous equations with an equal number of unknown coefficients ܷঋǡȰঌ and ܹ঍. The generalised eigenvalue problem for thermal buckling can be written as

(3.34) ቂሾࡷሿ ൅ߣሾࡷࢀሿቃ ሾઢሿ ൌͲǤ

It is noted that [ࡷ] and [ࡷࢀ] are the stiffness matrix and the coefficient matrix of temperature change respectively, and the vector ઢ is the eigenvector of the unknown coefficients from the polynomial trial functions. ߣҧ is the thermal buckling parameter which is equivalent to the critical temperature ȟܶ௖௥. To calculate a set of thermal buckling, the determinant of the coefficient matrix in Eq. (3.34) is set to zero.

Two types of solutions are taken into account in this paper, which are the temperature- independent material properties solution (called the Solution I) and the temperature- dependent material properties solution (the Solution II). To obtain the thermal buckling

Chapter 3

results of the Solution II, an iterative procedure is required that can be explained as the following steps.

(i) Beginning with the material properties of FG beams at the free stress

temperature ܶൌܶ଴ and then solving Eq. (3.34) for the critical temperature ȟܶ௖௥ which is the result of the Solution I.

(ii) Using a new value of temperature ܶൌܶ଴ ൅ȟܶ௖௥ to recalculate the coefficient matrix in Eq. (3.34) which provides a new critical temperature. (iii) Repeating step (ii) until reaching a given error tolerance (ߜ) as,

௜ାଵ ௜ ȟܶ௖௥ െȟܶ௖௥ (3.35) ߜൌቤ ௜ ቤ ൑ ͲǤͲͷΨǤ ȟܶ௖௥

In the case of vibration analysis including thermal effects, this can be achieved by multiplying the displacement and rotation functions of Eq. (3.31) with ݁௜ఠ௧ as the harmonic vibration, in which ߱ is the natural frequency. Substituting them into Eq. (3.30) and then repeating the procedure of minimisation as the thermal buckling analysis, the generalised eigenvalue problem for the vibration analysis including thermal effects can be written as

ଶ ൣሾࡷሿ ൅ ሾࡷࢀሿ൧ሾઢሿ െ߱ ሾࡹሿሾઢሿ ൌͲ (3.36) in which ሾࡹሿ is mass matrix. It is remarkable that for the Solution II of the vibration analysis the iterative procedure is not necessary because the value of temperature change is set as a given parameter in the thermo-elastic material properties Eq. (4.4) and the potential energy equation Eq. (3.22).

Chapter 3

3.2 (b) Thermal buckling and elastic vibration analysis of FG plates

The displacement field based on the improved TSDT that is used to describe the behaviour of FG plates in this research can be expressed as follows,

(ͷ Ͷ ͳ ͷ ߲ݓ (3.37a ݑൌݑ ሺݔǡ ݕǡ ݐሻ ൅ ൬ݖ െ ݖଷ൰߶ ሺݔǡ ݕǡ ݐሻ ൅൬ ݖെ ݖଷ൰ ଴ ଴ Ͷ ͵݄ଶ ௫ Ͷ ͵݄ଶ ߲ݔ

(ͷ Ͷ ଷ ͳ ͷ ଷ ߲ݓ଴ (3.37b ݒൌݒ଴ሺݔǡ ݕǡ ݐሻ ൅ ൬ݖ െ ଶ ݖ ൰߶௬ሺݔǡ ݕǡ ݐሻ ൅൬ ݖെ ଶ ݖ ൰ Ͷ ͵݄ Ͷ ͵݄ ߲ݕ

(ݓൌݓ଴ሺݔǡ ݕǡ ݐሻǤ (3.37c

To consider thermal buckling and elastic vibration of FG plates, the displacement field based on the improved TSDT (Shi, 2007) in Eq. (3.37) is used to find out strain components of the FG plates. The strain components can be represented in the form of normal and shear strains as follows.

߳௫௫ ሺ଴ሻ ሺଶሻ (3.38) ߛ௬௭ ߛ௬௭ ߛ௬௭ ൝߳௬௬ൡൌ൛߳ሺ଴ሻൟ൅ݖ൛߳ሺଵሻൟ൅ݖଷ൛߳ሺଷሻൟǡቄ ቅൌ൝ ൡ൅ݖଶ ൝ ൡǤ ߛ௫௭ ሺ଴ሻ ሺଶሻ ߛ௫௬ ߛ௫௭ ߛ௫௭

For the normal strain components:

߳ሺ଴ሻ ௫௫ ۗ ݑ଴ǡ௫ ۓ ሺ଴ሻ ሺ଴ሻ ݒ ൛߳ ൟൌ ߳௬௬ ൌ൝ ଴ǡ௬ ൡ (3.39) ሺ଴ሻۘ ݑ଴ǡ௬ ൅ݒ଴ǡ௫ ۔ ۙ ߛ௫௬ە

ͳ ۗ ൫ͷ߶௫ǡ௫ ൅ݓǡ௫௫൯ ۓ ሺଵሻ ۖ ߳௫௫ ۗ ۖ Ͷۓ ͳ ሺଵሻ ߳ሺଵሻ (൛߳ ൟൌ ௬௬ ൌ ൫ͷ߶௬ǡ௬ ൅ݓǡ௬௬൯ (3.40 ۘ Ͷ ۔ ሺଵሻۘ ۔ ۖ ߛ௫௬ ۙ ۖͳە ൫ͷ߶ ൅ʹݓ ൅ͷ߶ ൯ ۙ Ͷ ௫ǡ௬ ǡ௫௬ ௬ǡ௫ە

Chapter 3

െͷ ۗ ଶ ൫߶௫ǡ௫ ൅ݓǡ௫௫൯ ۓ ሺଷሻ ۖ ݄͵ ۖ ۗ ߳௫௫ۓ ሺଷሻ െͷ ൛߳ሺଷሻൟൌ ߳ ൌ (௬௬ ଶ ൫߶௬ǡ௬ ൅ݓǡ௬௬൯ (3.41 ۘ ݄͵ ۔ ሺଷሻۘ ۔ ۖ ߛ௫௬ ۙ ۖ െͷە ൫߶ ൅ʹݓ ൅߶ ൯ ۙ ଶ ௫ǡ௬ ǡ௫௬ ௬ǡ௫݄͵ە

For the shear strain components:

ͷ ሺ଴ሻ ൫߶௬ ൅ݓǡ௬൯ ߛ௬௭ ൝ ൡൌ൞Ͷ ൢ (3.42) ሺ଴ሻ ͷ ߛ௫௭ ൫߶ ൅ݓ ൯ Ͷ ௫ ǡ௫

െͷ ሺଶሻ ൫߶௬ ൅ݓǡ௬൯ ߛ௬௭ ଶ ൝ ൡൌ൞݄ ൢ (3.43) ሺଶሻ െͷ ߛ௫௭ ൫߶ ൅ݓ ൯ ݄ଶ ௫ ǡ௫

In plate analysis, it is more convenient to use a prime (,) as the partial differentiation with డ௨ respect to x and y coordinates. For example, ݑ is బ. The elastic constants of the ଴ǡ௫ డ௫ constitutive equations of FG plate in Eq. (3.1) are

(ሺݖǡ ܶሻ (3.44aܧሺݖǡ ܶሻ ߥܧ ܳ ൌܳ ൌ ǡܳ ൌܳ ൌ ǡ ଵଵ ଶଶ ͳെߥଶ ଵଶ ଶଵ ͳെߥଶ

(ሺݖǡ ܶሻ (3.44bܧ ሺݖሻ ൌ Ǥܩൌܳ ൌܳ ൌ ܳ ସସ ହହ ଺଺ ʹሼͳ൅ߥሽ

Chapter 3

3.2.1 Strain energy for FG plates

The total strain energy of a plate due to the normal force, shear force, moment resultants and the higher order terms can be expressed as:

௕ ௔ ͳ ܷ ൌ නනቂܰ ߳ሺ଴ሻ ൅ܰ ߳ሺ଴ሻ ൅ܰ ߛሺ଴ሻ ൅ܯ ߳ሺଵሻ ൅ܯ ߳ሺଵሻ ൅ܯ ߛሺଵሻ ௘ ʹ ௫௫ ௫௫ ௬௬ ௬௬ ௫௬ ௫௬ ௫௫ ௫௫ ௬௬ ௬௬ ௫௬ ௫௬ (3.45) ଴ ଴ ሺଷሻ ሺଷሻ ሺଷሻ ሺ଴ሻ ሺ଴ሻ ሺଶሻ ൅ܲ௫௫߳௫௫ ൅ܲ௬௬߳௬௬ ൅ܲ௫௬ߛ௫௬ ൅ܳ௫ߛ௫௭ ൅ܳ௬ߛ௬௭ ൅ܴ௫ߛ௫௭ ሺଶሻ ൅ܴ௬ߛ௬௭ ቃ݀ݔ݀ݕǤ

where

ܰ௫௫ ߪ ௫௫ ۗ ۓ ۗ ߪ ۓ ܰ ௬௬ ۖ ௛ଶΤ ௬௬ ܲ ௛ଶΤ ߪ ۖ ߬ ۖ ௫௫ ௫௫ ۖ ۖ ܰۖ ௫௬ ௫௬ ܲ ଷ ߪ (ൌන ݖߪ ݀ݖǡ ቐ ௬௬ቑ ൌනݖ൝ ௬௬ൡ ݀ݖǡ (3.46 ௫௫ ۘ ߬ ۔ ۘ ௫௫ܯ۔ ௛Τ ଶ ݖߪ ܲ௫௬ ି௛Τ ଶ ௫௬ି ௬௬ۖ ۖ ௬௬ۖܯۖ ۙ ݖ߬ە ௫௬ۙ ௫௬ܯە

௛ଶΤ ௛ଶΤ ߬ ߬ ܳ௫ ௫௭ ܴ௫ ଶ ௫௭ (൜ ൠൌ න ቄ ቅ ݀ݖǡ ൜ ൠൌ න ݖ ቄ ቅ݀ݖǤ (3.47 ܳ௬ ߬௬௭ ܴ௬ ߬௬௭ ି௛Τ ଶ ି௛Τ ଶ

On substitution Eq. (3.1) into Eqs. (3.46-3.47), this can yield the following forms.

ሼܰሽ ൌ ሾܣሿ൛߳ሺ଴ሻൟ൅ሾܤሿ൛߳ሺଵሻൟ൅ሾܧሿ൛߳ሺଷሻൟ (3.48)

ሼܯሽ ൌ ሾܤሿ൛߳ሺ଴ሻൟ൅ሾܦሿ൛߳ሺଵሻൟ൅ሾܨሿ൛߳ሺଷሻൟ (3.49)

ሼܲሽ ൌ ሾܧሿ൛߳ሺ଴ሻൟ൅ሾܨሿ൛߳ሺଵሻൟ൅ሾܪሿ൛߳ሺଷሻൟ (3.50)

Chapter 3

ሺ଴ሻ ሺଶሻ (3.51) ܳ௫ ܣ଺଺ Ͳ ߛ௫௭ ܦ଺଺ Ͳ ߛ௫௭ ൜ ൠൌ൤ ൨൝ ሺ଴ሻൡ൅൤ ൨൝ ሺଶሻൡ ܳ௬ Ͳܣ Ͳܦ ଺଺ ߛ௬௭ ଺଺ ߛ௬௭

ሺ଴ሻ ሺଶሻ (3.52) ܴ௫ ܦ଺଺ Ͳ ߛ௫௭ ܨ଺଺ Ͳ ߛ௫௭ ൜ ൠൌ൤ ൨൝ ሺ଴ሻൡ൅൤ ൨൝ ሺଶሻൡ ܴ௬ Ͳܦ Ͳܨ ଺଺ ߛ௬௭ ଺଺ ߛ௬௭

The material stiffness components can be defined as the extensional, bending-extensional, bending, warping-extensional, warping-bending and warping-higher order bending stiffness components; ܣ௜௝ǡܤ௜௝ǡܦ௜௝ǡܧ௜௝ǡܨ௜௝ǡܪ௜௝ǡሺ݅ǡ݆ൌͳǡʹǡ͸ሻ. All of the stiffness components are expressed as:

௛ଶΤ ଶ ଷ ସ ଺ (௜௝൯ൌ න ܳ௜௝ሺͳǡ ݖǡ ݖ ǡݖ ǡݖ ǡݖ ሻ ݀ݖǤ (3.53ܪ௜௝ǡܨ௜௝ǡܧ௜௝ǡܦ௜௝ǡܤ௜௝ǡܣ൫ ି௛Τ ଶ

The material stiffness components in the form of the material volume fraction index (n) for FG plate analysis are derived by using integration by parts and substitution techniques, which can be presented as,

݄ ܧ (3.54a) ܣ ൌܣ ൌ ൤ ௧௕ ൅ܧ ൨ ଵଵ ଶଶ ͳെߥଶ ሺ݊൅ͳሻ ௕

ሺͳ െ ߥሻܣ (3.54b) ܣ ൌܣ ൌߥܣ ǡܣൌ ଵଵ ଵଶ ଶଵ ଵଵ ଺଺ ʹ

ܧ ݄ଶ ݊ (3.54c) ܤ ൌ ௧௕ ൤ ൨ ଵଵ ͳെߥଶ ʹሺ݊൅ͳሻሺ݊൅ʹሻ

ሺͳ െ ߥሻܤଵଵ (3.54d) ܤ ൌܤ ൌߥܤ ǡܤൌ ଵଶ ଶଵ ଵଵ ଺଺ ʹ

ଷ ଶ ݄ ܧ௧௕ሺ݊ ൅݊൅ʹሻ ܧ௕ (3.54e) ܦ ൌ ቈ ൅ ቉ ଵଵ ͳെߥଶ Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻ ͳʹ

Chapter 3

ሺͳ െ ߥሻܦଵଵ (3.54f) ܦ ൌܦ ൌߥܦ ǡܦൌ ଵଶ ଶଵ ଵଵ ଺଺ ʹ

ସ ଶ ܧ௧௕݄ ݊ሺ݊ ൅͵݊൅ͺሻ (3.54g) ܧ ൌ ቈ ቉ ଵଵ ͳെߥଶ ͺሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻ

ሺͳ െ ߥሻܧଵଵ (3.54h) ܧ ൌܧ ൌߥܧ ǡܧൌ ଵଶ ଶଵ ଵଵ ଺଺ ʹ

ହ ସ ଷ ଶ ݄ ܧ௧௕ሺ݊ ൅͸݊ ൅ ʹ͵݊ ൅ ͳͺ݊ ൅ ʹͶሻ ܧ௕ (3.54i) ܨ ൌ ቈ ൅ ቉ ଵଵ ͳെߥଶ ͳ͸ሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻ ͺͲ

ሺͳ െ ߥሻܨଵଵ (3.54j) ܨ ൌܨ ൌߥܨ ǡܨൌ ଵଶ ଶଵ ଵଵ ଺଺ ʹ

݄଻ ܧ ሺ݊଺ ൅ͳͷ݊ହ ൅ ͳͳͷ݊ସ ൅ ͶͲͷ݊ଷ ൅ ͻ͸Ͷ݊ଶ ൅ ͸͸Ͳ݊ ൅ ͹ʹͲሻ (3.54k) ܪ ൌ ቈ ௧௕ ଵଵ ͳെߥଶ ͸Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻሺ݊൅͸ሻሺ݊൅͹ሻ ܧ ൅ ௕ ቉ ͶͶͺ

ሺͳ െ ߥሻܪ (3.54l) ܪ ൌܪ ൌߥܪ ǡܪൌ ଵଵ ଵଶ ଶଵ ଵଵ ଺଺ ʹ

The strain energy equation of FG plates as functions of material stiffness and strain components can be written as,

Chapter 3

௕ ௔ ͳ మ మ ܷ ൌ නනቂ൬ܣ ߳ሺ଴ሻ ൅ʹܣ ߳ሺ଴ሻ߳ሺ଴ሻ ൅ܣ ߳ሺ଴ሻ ௘ ʹ ଵଵ ௫௫ ଵଶ ௫௫ ௬௬ ଶଶ ௬௬ ଴ ଴ ሺ଴ሻమ ሺ଴ሻమ ሺ଴ሻమ ൅ܣ଺଺ ቀߛ௫௬ ൅ߛ௫௭ ൅ߛ௬௭ ቁ൰ ሺ଴ሻ ሺଵሻ ሺ଴ሻ ሺଵሻ ሺ଴ሻ ሺଵሻ ሺ଴ሻ ሺଵሻ ൅ʹቀܤଵଵ߳ ߳ ൅ܤଵଶቀ߳ ߳ ൅߳ ߳ ቁ൅ܤଶଶ߳ ߳ ௫௫ ௫௫ ௫௫ ௬௬ ௬௬ ௫௫ ௬௬ ௬௬ ൅ܤ ߛሺ଴ሻߛሺଵሻቁ ଺଺ ௫௬ ௫௬ (3.55) ሺଵሻమ ሺଵሻ ሺଵሻ ሺଵሻమ ൅൬ܦଵଵ߳௫௫ ൅ʹܦଵଶ߳௫௫ ߳௬௬ ൅ܦଶଶ߳௬௬ ሺଵሻమ ሺ଴ሻ ሺଶሻ ሺ଴ሻ ሺଶሻ ൅ܦ଺଺ ቀߛ௫௬ ൅ʹߛ௫௭ ߛ௫௭ ൅ʹߛ௬௭ ߛ௬௭ ቁ൰ ሺ଴ሻ ሺଷሻ ሺ଴ሻ ሺଷሻ ሺ଴ሻ ሺଷሻ ሺ଴ሻ ሺଷሻ ൅ʹቀܧଵଵ߳௫௫ ߳௫௫ ൅ܧଵଶቀ߳௫௫ ߳௬௬ ൅߳௬௬ ߳௫௫ ቁ൅ܧଶଶ߳௬௬ ߳௬௬ ሺ଴ሻ ሺଷሻ ൅ܧ଺଺ߛ௫௬ ߛ௫௬ ቁ ሺଵሻ ሺଷሻ ሺଵሻ ሺଷሻ ሺଵሻ ሺଷሻ ሺଵሻ ሺଷሻ ൅ʹቀܨଵଵ߳௫௫ ߳௫௫ ൅ܨଵଶቀ߳௫௫ ߳௬௬ ൅߳௬௬ ߳௫௫ ቁ൅ܨଶଶ߳௬௬ ߳௬௬ ቁ ሺଵሻ ሺଷሻ ሺଶሻమ ሺଶሻమ ൅ܨ଺଺ ቀʹߛ௫௬ ߛ௫௬ ൅ߛ௬௭ ൅ߛ௫௭ ቁ ሺଷሻమ ሺଷሻ ሺଷሻ ሺଷሻమ ሺଷሻమ ଺଺ߛ௫௬ ቁቃ ݀ݔ݀ݕǤܪଶଶ߳௬௬ ൅ܪଵଶ߳௫௫ ߳௬௬ ൅ܪʹଵଵ߳௫௫ ൅ܪ൅ቀ

3.1.2 The potential energy for FG plates due to thermal stresses

For a typical FG plate which has been in a high temperature environment for a long period of time, it is assumed that the temperature can distribute uniformly across its thickness so that the case of uniform temperature rise is taken into consideration in this investigation.

The initial temperature is ܶ଴ (ܶ଴ ൌ ͵ͲͲܭሻ , which is a stress free state, and then changes to final temperature with ȟܶ . Hence, the thermal stresses occur within the FG plate as

் ߪ௫௫ ܳଵଵ ܳଵଶ Ͳ ߙሺݖǡ ܶሻ ் (ቐߪ௬௬ቑൌെ൥ܳଶଵ ܳଶଶ Ͳ ൩൝ߙሺݖǡ ܶሻൡȟܶǤ (3.56 ் ͲͲܳ ߬௫௬ ଺଺ Ͳ

The thermal stresses in terms of normal stresses are considered with the increase in temperature; while, thermal shear stress is negligible.

(ሺݖǡ ܶሻߙሺݖǡ ܶሻȟܶ (3.57ܧ ߪ் ൌߪ் ൌߪ் ൌെ ǡ்߬ ൌͲ ௫௫ ௬௬ ͳെߥ ௫௬

Chapter 3

The potential energy due to thermal stress can be expressed as follows (Kim, 2005),

ͳ ் ் ் (3.58) ௘ ൌ ׬ ൣߪ௫௫݀௫௫ ൅ʹߪ௫௬݀௫௬ ൅ߪ௬௬݀௬௬൧ܸ݀ǡܸ ʹ ௏

(௜௝ ൌݑǡ௜ݑǡ௝ ൅ݒǡ௜ݒǡ௝ ൅ݓǡ௜ݓǡ௝ሺ݅ǡ ݆ ൌ ݔǡ ݕሻǤ (3.59݀

The potential energy equation, which is written in terms of material stiffness and normal strain components in relation to thermal stresses, can be expressed as the following equation.

௕ ௔ ͳ ܸ ൌ නනቂܣ் ቀȩሺ଴ሻ ൅ȩሺ଴ሻ ൅ȩሺ଴ሻቁ൅ܤ் ቀȩሺଵሻ ൅ȩሺଵሻቁ൅ܦ் ቀȩሺଶሻ ൅ȩሺଶሻቁ ௘ ʹ ଵଵ ௨௫ ௩௫ ௪௫ ଵଵ ௨௫ ௩௫ ଵଵ ௨௫ ௩௫ ଴ ଴ ் ሺଷሻ ሺଷሻ ் ሺସሻ ሺସሻ ் ሺ଺ሻ ሺ଺ሻ (3.60) ൅ܧଵଵቀȩ௨௫ ൅ȩ௩௫ ቁ൅ܨଵଵቀȩ௨௫ ൅ȩ௩௫ ቁ൅ܪଵଵቀȩ௨௫ ൅ȩ௩௫ ቁ ் ሺ଴ሻ ሺ଴ሻ ሺ଴ሻ ் ሺଵሻ ሺଵሻ ൅ܣଶଶቀȩ௨௬ ൅ȩ௩௬ ൅ȩ௪௬ቁ൅ܤଶଶቀȩ௨௬ ൅ȩ௩௬ ቁ ் ሺଶሻ ሺଶሻ ் ሺଷሻ ሺଷሻ ் ሺସሻ ሺସሻ ൅ܦଶଶቀȩ௨௬ ൅ȩ௩௬ ቁ൅ܧଶଶቀȩ௨௬ ൅ȩ௩௬ ቁ൅ܨଶଶቀȩ௨௬ ൅ȩ௩௬ ቁ ் ሺ଺ሻ ሺ଺ሻ ଶଶቀȩ௨௬ ൅ȩ௩௬ ቁቃ ݀ݔ݀ݕǤܪ൅

The normal strain components due to thermal stresses in Eq. (3.60) are presented in the following forms,

ሺ଴ሻ ଶ ሺ଴ሻ ଶ ሺ଴ሻ ଶ (3.61a) ȩ௨క ൌݑ଴ǡకǢȩ௩క ൌݒ଴ǡకǢȩ௪క ൌݓǡక

ͳ ͳ (3.61b) ȩሺଵሻ ൌ ൫ͷݑ ߶ ൅ݑ ݓ ൯Ǣȩሺଵሻ ൌ ൫ͷݒ ߶ ൅ݒ ݓ ൯ ௨క ʹ ଴ǡక ௫ǡక ଴ǡక ǡ௫క ௩క ʹ ଴ǡక ௬ǡక ଴ǡక ǡ௬క

ͳ (3.61c) ȩሺଶሻ ൌ ൫ʹͷ߶ଶ ൅ͳͲ߶ ݓ ൅ݓଶ ൯ ௨క ͳ͸ ௫ǡక ௫ǡక ǡ௫క ǡ௫క

ͳ (3.61d) ȩሺଶሻ ൌ ൫ʹͷ߶ଶ ൅ͳͲ߶ ݓ ൅ݓଶ ൯ ௩క ͳ͸ ௬ǡక ௬ǡక ǡ௬క ǡ௬క

Chapter 3

െͳͲ െͳͲ (3.61e) ȩሺଷሻ ൌ ൫ݑ ߶ ൅ݑ ݓ ൯Ǣȩሺଷሻ ൌ ൫ݒ ߶ ൅ݒ ݓ ൯ ௨క ͵݄ଶ ଴ǡక ௫ǡక ଴ǡక ǡ௫క ௩క ͵݄ଶ ଴ǡక ௬ǡక ଴ǡక ǡ௬క

െͳͲ (3.61f) ȩሺସሻ ൌ ൫ͷ߶ଶ ൅͸߶ ݓ ൅ݓଶ ൯ ௨క ͳʹ݄ଶ ௫ǡక ௫ǡక ǡ௫క ǡ௫క

െͳͲ (3.61g) ȩሺସሻ ൌ ൫ͷ߶ଶ ൅͸߶ ݓ ൅ݓଶ ൯ ௩క ͳʹ݄ଶ ௬ǡక ௬ǡక ǡ௬క ǡ௬క

ʹͷ (3.61h) ȩሺ଺ሻ ൌ ൫߶ଶ ൅ʹ߶ ݓ ൅ݓଶ ൯ ௨క ͻ݄ସ ௫ǡక ௫ǡక ǡ௫క ǡ௫క

ʹͷ (3.61i) ȩሺ଺ሻ ൌ ൫߶ଶ ൅ʹ߶ ݓ ൅ݓଶ ൯ǡሺߦൌݔǡݕሻǤ ௩క ͻ݄ସ ௬ǡక ௬ǡక ǡ௬క ǡ௬క

The thermal stresses and their higher-order terms are

௛ଶΤ ் ் ் ் ் ் ் ଶ ଷ ସ ଺ (௜௜ሻ ൌනߪ௜௜ ሺͳǡ ݖǡ ݖ ǡݖ ǡݖ ǡݖ ሻ ݀ݖǡሺ݅ൌͳǡʹሻǤ (3.62ܪ௜௜ ǡܨ௜௜ǡܧ௜௜ǡܦ௜௜ǡܤ௜௜ǡܣሺ ି௛Τ ଶ Similarly, the material stiffness components associated with thermal stresses in Eq. (3.62) can also be presented in another form as:

݄οܶ ܧ௧௕ߙ௧௕ ܧ௧௕ߙ௕ ൅ܧ௕ߙ௧௕ (3.63a) ܣ் ൌܣ் ൌ ൤ ൅ ൅ܧ ߙ ൨ ଵଵ ଶଶ ͳെߥ ሺʹ݊ ൅ ͳሻ ሺ݊ ൅ ͳሻ ௕ ௕

݄݊ଶοܶ ܧ ߙ ܧ ߙ ൅ܧ ߙ (3.63b) ܤ் ൌܤ் ൌ ൤ ௧௕ ௧௕ ൅ ௧௕ ௕ ௕ ௧௕ ൨ ଵଵ ଶଶ ͳെߥ ʹሺ݊ ൅ ͳሻሺʹ݊ ൅ ͳሻ ʹሺ݊൅ͳሻሺ݊ ൅ ʹሻ

ଷ ଶ ் ் ݄ οܶ ܧ௧௕ߙ௧௕ሺʹ݊ ൅݊൅ͳሻ ܦଵଵ ൌܦଶଶ ൌ ቈ ͳെߥ ʹሺʹ݊ ൅ ͳሻሺʹ݊ ൅ ʹሻሺʹ݊ ൅ ͵ሻ (3.63c) ሺܧ ߙ ൅ܧ ߙ ሻሺ݊ଶ ൅݊൅ʹሻ ܧ ߙ ൅ ௧௕ ௕ ௕ ௧௕ ൅ ௕ ௕቉ Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊ ൅ ͵ሻ ͳʹ

Chapter 3

ସ ଶ ் ் ݄݊ οܶ ܧ௧௕ߙ௧௕ሺʹ݊ ൅͵݊൅Ͷሻ ܧଵଵ ൌܧଶଶ ൌ ቈ ͳെߥ ʹሺʹ݊ ൅ ͳሻሺʹ݊ ൅ ʹሻሺʹ݊ ൅ ͵ሻሺʹ݊ ൅ Ͷሻ (3.63d) ሺܧ ߙ ൅ܧ ߙ ሻሺ݊ଶ ൅͵݊൅ͺሻ ൅ ௧௕ ௕ ௕ ௧௕ ቉ ͺሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊ ൅ Ͷሻ

ହ ସ ଷ ଶ ் ் ݄ οܶ ܧ௧௕ߙ௧௕ሺͶ݊ ൅ͳʹ݊ ൅ʹ͵݊ ൅ͻ݊൅͸ሻ ܨଵଵ ൌܨଶଶ ൌ ቈ ͳെߥ Ͷሺʹ݊ ൅ ͳሻሺʹ݊ ൅ ʹሻሺʹ݊ ൅ ͵ሻሺʹ݊ ൅ Ͷሻሺʹ݊ ൅ ͷሻ (3.63e) ሺܧ ߙ ൅ܧ ߙ ሻሺ݊ସ ൅͸݊ଷ ൅ ʹ͵݊ଶ ൅ ͳͺ݊ ൅ ʹͶሻ ܧ ߙ ൅ ௧௕ ௕ ௕ ௧௕ ൅ ௕ ௕቉ ͳ͸ሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊ ൅ ͷሻ ͺͲ

் ் ܪଵଵ ൌܪଶଶ ݄଻οܶ ܧ ߙ ሺͺ݊଺ ൅͸Ͳ݊ହ ൅ ʹ͵Ͳ݊ସ ൅ ͶͲͷ݊ଷ ൅ Ͷͺʹ݊ଶ ൅ ͳ͸ͷ݊ ൅ ͻͲሻ ൌ ቈ ௧௕ ௧௕ ͳെߥ ͺሺʹ݊ ൅ ͳሻሺʹ݊ ൅ ʹሻሺʹ݊ ൅ ͵ሻሺʹ݊ ൅ Ͷሻሺʹ݊ ൅ ͷሻሺʹ݊ ൅ ͸ሻሺʹ݊ ൅ ͹ሻ ሺܧ ߙ ൅ܧ ߙ ሻሺ݊଺ ൅ͳͷ݊ହ ൅ ͳͳͷ݊ସ ൅ ͶͲͷ݊ଷ ൅ ͻ͸Ͷ݊ଶ ൅ ͸͸Ͳ݊ ൅ ͹ʹͲሻ (3.63f) ൅ ௧௕ ௕ ௕ ௧௕ ͸Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻሺ݊൅͸ሻሺ݊ ൅ ͹ሻ ܧ ߙ ൅ ௕ ௕቉ ͶͶͺ

3.2.3 The kinetic energy for FG plates

The kinetic energy which is required for analysing vibration of an FG plate,

ͳ ଶ ଶ ଶ (3.64) ௘ ൌ ׬ ߩሺݖǡ ܶሻሾݑሶ ൅ݒሶ ൅ݓሶ ሿܸ݀ǡܶ ʹ ௏ where dot is defined as the differentiation with respect to time (t). The displacement functions in Eq. (3.37) are substituted into Eq. (3.64) to produce the kinetic energy equation based on the TSDT which can be expressed as the following equation,

Chapter 3

௕ ௔ ͳ ͳ ܫ൅ ൫ͷ൫ݑሶ ߶ሶ ൅ݒሶ ߶ሶ ൯൅ݑሶ ݓሶ ൅ݒሶ ݓሶ ൯ ܫൌ නන൤ሺݑሶ ଶ ൅ݒሶ ଶ ൅ݓሶ ଶሻ ܶ ௘ ʹ ଴ ଴ ଴ ʹ ଴ ௫ ଴ ௬ ଴ ǡ௫ ଴ ǡ௬ ଵ ଴ ଴

ͳ ܫ൅ ൫ݓሶ ଶ ൅ʹͷ൫߶ሶ ଶ ൅߶ሶ ଶ൯൅ͳͲ൫߶ሶ ݓሶ ൅߶ሶ ݓሶ ൯൅ݓሶ ଶ ൯ ͳ͸ ǡ௫ ௫ ௬ ௫ ǡ௫ ௬ ǡ௬ ǡ௬ ଶ ͳͲ (3.65) ܫെ ൫ݑሶ ߶ሶ ൅ݑሶ ݓሶ ൅ݒሶ ߶ሶ ൅ݒሶ ݓሶ ൯ ͵݄ଶ ଴ ௫ ଴ ǡ௫ ଴ ௬ ଴ ǡ௬ ଷ ͳͲ ܫെ ൫ݓሶ ଶ ൅ͷ൫߶ሶ ଶ ൅߶ሶ ଶ൯൅͸൫߶ሶ ݓሶ ൅߶ሶ ݓሶ ൯൅ݓሶ ଶ ൯ ͳʹ݄ଶ ǡ௫ ௫ ௬ ௫ ǡ௫ ௬ ǡ௬ ǡ௬ ସ ʹͷ ൨ ݀ݔ݀ݕǡ ܫ൅ ൫ݓሶ ଶ ൅߶ሶ ଶ ൅ʹ൫߶ሶ ݓሶ ൅߶ሶ ݓሶ ൯൅߶ሶ ଶ ൅ݓሶ ଶ ൯ ͻ݄ସ ǡ௫ ௫ ௫ ǡ௫ ௬ ǡ௬ ௬ ǡ௬ ଺

௛ଶΤ ൌ ߩሺݖǡ ܶሻݖ௜݀ݖ ǡ݅ ൌ Ͳǡͳǡʹǡ͵ǡͶǡ͸Ǥ ܫ where ௜ ׬ି௛ȀΤ ଶ

It is noted that the components of moment of inertia ሺܫ௜ሻ as the function of the power law or volume fraction index (n) are expressed as follows,

ߩ௧௕ (3.66a) ܫ ൌ݄൤ ൅ߩ ൨ ଴ ሺ݊൅ͳሻ ௕

݊ (3.66b) ܫ ൌ݄ଶߩ ൤ ൨ ଵ ௧௕ ʹሺ݊൅ͳሻሺ݊൅ʹሻ

ߩ ሺ݊ଶ ൅݊൅ʹሻ ߩ (3.66c) ܫ ൌ݄ଷ ቈ ௧௕ ൅ ௕ ቉ ଶ Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻ ͳʹ

݊ሺ݊ଶ ൅͵݊൅ͺሻ (3.66d) ܫ ൌ݄ସߩ ቈ ቉ ଷ ௧௕ ͺሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻ

ߩ ሺ݊ସ ൅͸݊ଷ ൅ʹ͵݊ଶ ൅ ͳͺ݊ ൅ ʹͶሻ ߩ (3.66e) ܫ ൌ݄ହ ቈ ௧௕ ൅ ௕ ቉ ସ ͳ͸ሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻ ͺͲ

ߩ ሺ݊଺ ൅ ͳͷ݊ହ ൅ ͳͳͷ݊ସ ൅ ͶͲͷ݊ଷ ൅ ͻ͸Ͷ݊ଶ ൅ ͸͸Ͳ݊ ൅ ͹ʹͲሻ ߩ (3.66f) ܫ ൌ݄଻ ቈ ௧௕ ൅ ௕ ቉ ଺ ͸Ͷሺ݊൅ͳሻሺ݊൅ʹሻሺ݊൅͵ሻሺ݊൅Ͷሻሺ݊൅ͷሻሺ݊൅͸ሻሺ݊൅͹ሻ ͶͶͺ

Chapter 3

The total energy functional ሺȫሻ of FG plates for the thermal buckling analysis can then be written as the following:

ȫൌܷ௘ ൅ܸ௘Ǥ (3.67)

For the case of vibration analysis including thermal effects, the total energy functional ሺȫሻ of FG plates is

ȫൌܷ௘ ൅ܸ௘ െܶ௘Ǥ (3.68)

3.2.4 Forced vibration analysis of FG plates

௜ఠ೐ೣ௧ Considering an FG plate subjected to a uniformly distributed dynamic force ሺܲ଴݁ ሻ over the plate domain of dimensions a and b, work done by the external force can be expressed as

௕ ௔ (3.69) ௜ఠ೐ೣ௧ ௘ ൌܲ଴݁ නනݓ݀ݔ݀ݕǤܹ ଴ ଴

It is noted that ߱௘௫ is the frequency of external excitation. For the case of forced vibration analysis at high temperature environment, the total energy functional includes the sum of the strain energy, the energy due to applied load and thermal load and the kinetic energy, which can be written as follows:

ȫൌܷ௘ ൅ܹ௘ ൅ܸ௘ െܶ௘Ǥ (3.70)

3.2.5 The solution method for FG plate analysis

The governing equation or the total energy functional based on the improved TSDT for FG plate analysis derived from the energy approach can be solved using the Ritz method in order to determine the thermal buckling and natural frequency results. Two types of simply supported boundary conditions, which are a movable boundary condition denoted by SS-1 and an immovable boundary condition denoted by SS-2, are chosen to consider the plate analysis in this research.

Chapter 3

For SS-1: the essential and natural conditions are

௫௫ ൌͲǡሺݔൌͲǡܽሻܯݒ଴ǡݓǡ߶௬ǡܰ௫௫ǡ

௬௬ ൌͲǡሺݕൌͲǡܾሻǤܯݑ଴ǡݓǡ߶௫ǡܰ௬௬ǡ

The assumed displacement and rotation functions that satisfy the conditions of SS-1 are expanded in the double trigonometric (Fourier) series in terms of unknown parameters, which can be written in the following expansions:

ெ ே (3.71a) ߙ௠ݔߚ௡ݕ ݑ଴ሺݔǡ ݕሻ ൌ෍෍ܷ௠௡ ௠ୀଵ ௡ୀଵ

ெ ே (3.71b) ߚ௡ݕ ݒ଴ሺݔǡ ݕሻ ൌ෍෍ܸ௠௡ߙ௠ݔ ௠ୀଵ ௡ୀଵ

ெ ே (3.71c) ݓ଴ሺݔǡ ݕሻ ൌ෍෍ܹ௠௡ߙ௠ݔߚ௡ݕ ௠ୀଵ ௡ୀଵ

ெ ே (3.71d) ߙ௠ݔߚ௡ݕ ߶௫ሺݔǡ ݕሻ ൌ෍෍Ȱ௫௠௡ ௠ୀଵ ௡ୀଵ

ெ ே (3.71e) ߚ௡ݕ ߶௬ሺݔǡ ݕሻ ൌ෍෍Ȱ௬௠௡ߙ௠ݔ ௠ୀଵ ௡ୀଵ

For SS-2: the essential and natural conditions are:

௫௫ ൌͲǡሺݔൌͲǡܽሻܯݑ଴ǡݓǡ߶௬ǡܰ௫௬ǡ

௬௬ ൌͲǡሺݕൌͲǡܾሻǤܯݒ଴ǡݓǡ߶௫ǡܰ௫௬ǡ

The assumed in-plane displacements satisfying the conditions of SS-2 in direction of x and y are replaced by,

Chapter 3

ெ ே (3.72a) ߚ௡ݕ ݑ଴ሺݔǡ ݕሻ ൌ෍෍ܷ௠௡ߙ௠ݔ ௠ୀଵ ௡ୀଵ

ெ ே (3.72b) ߙ௠ݔߚ௡ݕ ݒ଴ሺݔǡ ݕሻ ൌ෍෍ܸ௠௡ ௠ୀଵ ௡ୀଵ

However, for the functions of ݓ଴ǡ߶௫ and ߶௬ of SS-2, the expressions can be represented the same as the case of SS-1.

To consider the plate which is supported by clamped boundary condition denoted by CC, the conditions of this fully clamped are,

ݑ଴ǡݒ଴ǡݓǡ߶௫ǡ߶௬ǡݓǡ௫ ൌͲǡሺݔൌͲǡܽሻ

ݑ଴ǡݒ଴ǡݓǡ߶௫ǡ߶௬ǡݓǡ௬ ൌͲǡሺݕൌͲǡܾሻ

The assumed displacement and rotation functions that can satisfy the fully clamped conditions are expressed as following functions (Ugural, 1999, Shi et al., 2004):

ெ ே (3.73a) ݑ଴ሺݔǡ ݕሻ ൌ෍෍ܷ௠௡ߙ௠ݔߚ௡ݕ ௠ୀଵ ௡ୀଵ

ெ ே (3.73b) ݒ଴ሺݔǡ ݕሻ ൌ෍෍ܸ௠௡ߙ௠ݔߚ௡ݕ ௠ୀଵ ௡ୀଵ

ெ ே (3.73c) ʹߚ௡ݕሻ ʹߙ௠ݔሻሺͳ െ ݓ଴ሺݔǡ ݕሻ ൌ෍෍ܹ௠௡ሺͳെ ௠ୀଵ ௡ୀଵ

ெ ே (3.73d) ߶௫ሺݔǡ ݕሻ ൌ෍෍Ȱ௫௠௡ߙ௠ݔߚ௡ݕ ௠ୀଵ ௡ୀଵ

ெ ே (3.73e) ߶௬ሺݔǡ ݕሻ ൌ෍෍Ȱ௬௠௡ߙ௠ݔߚ௡ݕ ௠ୀଵ ௡ୀଵ

Chapter 3

௠గ ௡గ where ߙ ൌ ǡߚ ൌ ǡ and ܷ ǡܸ ǡܹ ǡȰ ǡȰ are unknown coefficients. ௠ ௔ ௡ ௕ ௠௡ ௠௡ ௠௡ ௫௠௡ ௬௠௡

Similar to the beam analysis, for thermal buckling of FG plates, the assumed displacement and rotation functions corresponding to boundary conditions are substituted into the total energy functionalሺȫሻ of Eq. (3.67) and then taking derivative the functional with respect to the unknown coefficients in the procedure of minimisation.

This procedure leads to a system of simultaneous equations equal in number to the number of unknown coefficients ܷ௠௡ǡܸ௠௡ǡܹ௠௡ǡȰ௫௠௡ and Ȱ௬௠௡. The generalised eigenvalue problem for thermal buckling can be written as

(3.74) ቂሾࡷሿ ൅ߣሾࡷࢀሿቃ ሾઢሿ ൌͲǤ where [ࡷ] and [ࡷࢀ] are the stiffness matrix and the coefficient matrix of temperature change, respectively, and the vector ઢ is the eigenvector of the unknown coefficients. The parameter ߣҧ is the thermal buckling result which is equivalent to the critical temperatureሺȟܶ௖௥ሻ. To calculate a set of thermal buckling, the determinant of the coefficient matrix in Eq. (3.74) is set to zero. In this research, the thermal buckling results of FG plates related to temperature dependent solution are also investigated by using the simple iterative technique. The details of the iterative procedure are already described in the section of FG beam analysis, to obtain the thermal buckling results of the Solution II.

In terms of thermo-elastic vibration of FG plates, to obtain the results of free vibration analysis, the assumed displacement and rotation functions are multiplied with ݁௜ఠ௧ as the harmonic vibration. The parameter ߱ is the natural frequency of the plate associated with (m,n) mode. Substituting the assumed functions into the total energy functional of Eq. (3.68) and then repeating the procedure of minimisation, the generalised eigenvalue problem for the vibration analysis including thermal effects can be written as

ଶ ൣሾࡷሿ ൅ ሾࡷࢀሿ൧ሾઢሿ െ߱ ሾࡹሿሾઢሿ ൌͲ (3.75) where ሾࡹሿ is mass matrix. It is noted that for the Solution II of the vibration analysis, the iterative procedure is not necessary because the value of temperature change is set as a

Chapter 3

given parameter in the thermo-elastic material properties in Eq. (4.4) and the potential energy equation of Eq. (3.60).

For forced vibration analysis, the generalised equation is

ଶ ൣሾࡷሿ ൅ ሾࡷࢀሿ൧ሾઢሿ െ߱ ሾࡹሿሾઢሿ ൌ ሾࢌሿ (3.76) where f is the force vector.

Chapter 4 Thermal buckling and vibration analysis of FG beams and plates: Applications

In this chapter, the theoretical formulations based on the improved TSDT as presented in Chapter 3 are used to find out thermal buckling and vibration results associated with different boundary conditions, using the Ritz method. The main contents in this chapter are

4.1 Application of the improved TSDT to FG beam analysis

4.2 Application of the improved TSDT to FG plate analysis

4.1 Application of the improved TSDT to FG beam analysis

Third order shear deformation theories (TSDTs) represent a quadratic variation of the transverse shear strains and transverse shear stresses throughout the thickness using the expanded displacements up to the cubic term in the thickness coordinate. Unlike the classical and lower-order theories, TSDTs can represent the kinetics better and satisfy the zero traction boundary conditions on the top and bottom surfaces without using shear correction factors.

As the mechanical properties of FG beam and plate vary across their thickness with a nonlinear profile, using TSDT which accounts for the cubic variation of displacement throughout the thickness, it is more suitable to predict the FG beam and plate behaviour than using classical and other theories. In Chapter 3, the theoretical formulations based on the TSDTs were constructed to deal with thermal buckling and elastic vibration of FG 95

Chapter 4

beams and plates. Therefore, in this chapter, it is worth applying those formulations to investigate thermal bucking and vibration characteristics of FG beams and plates.

4.1.1 FG beam and material properties

An FG beam composed of ceramic-metal is considered in this section. The beam has length (L), width (b) and thickness (h). The material composition at the top surface (z=h/2) is assumed to be the ceramic-rich and it is varying continuously to the metal-rich surface at the opposite side (z=-h/2). The geometry of the FG beam is shown in Fig. 4.1. The power law distribution is used for the volume fraction of the ceramic (ܸ௖) and the metal (ܸ௠) as:

(ݖ ͳ ௡ (4.1a ܸ ൌ൬ ൅ ൰ ௖ ݄ ʹ

ܸ௠ ൌͳെܸ௖Ǥ (4.1b)

z Ceramic-rich

h/2 x

-h/2 Metal-rich

b L Fig. 4.1 Geometry of functionally graded beam

It is noted that the positive real number n ሺͲ൑݊൑λሻ is the power law or volume fraction index, and z is the distance from the mid-plane of the FG beam. The FG beam becomes a fully ceramic beam when n is set to be zero, whereas n=∞ indicates a fully metallic beam. The variation of ceramic volume fraction across the FG beam thickness with the different power law or material volume fraction index (n) is represented in Fig. 4.2. It can be seen that the area under the graph in each n is the percentage of ceramic used to make FG beams; while, the complementary area is the percentage of metal. Hence, the effective

Chapter 4

material properties of FG beams (P), like Young’s modulus (E), the coefficient of thermal expansion (α) and density (ρ), can then be expressed as,

ܲൌܲ௧ܸ௖ ൅ܲ௕ܸ௠Ǥ (4.2)

Fig. 4.2 Variation of the volume fraction across thickness direction of the FG beams

The subscripts t and b mean the material properties at the top and bottom beam surfaces, respectively. From the relationship between ceramic and metal using the above relationship, Young’s modulus ሺܧሻ, the coefficient of thermal expansion ሺߙሻ and density ሺߩሻ, which are used to calculate the material stiffness and the moment of inertia for FGMs, can be presented as:

(ݖ ͳ ௡ (4.3a ǡ ܧሻ ൬ ൅ ൰ ൅ ܧെ ܧሺݖሻ ൌ ሺܧ ௧ ௕ ݄ ʹ ௕

(ݖ ͳ ௡ (4.3b ߙሺݖሻ ൌ ሺߙ െߙ ሻ ൬ ൅ ൰ ൅ߙ ǡ ௧ ௕ ݄ ʹ ௕

Chapter 4

(ݖ ͳ ௡ (4.3c ߩሺݖሻ ൌ ሺߩ െߩ ሻ ൬ ൅ ൰ ൅ߩ ǡ ௧ ௕ ݄ ʹ ௕

However, Poisson’s ratio ሺߥሻ is assumed to be constant because there is a small difference in the Poisson’s ratio value between ceramic and metal compared to other material properties; therefore, the average value is used for computation.

The common use of FGMs in high temperature environments leads to considerable changes in material properties. For example, Young’s modulus usually decreases, and the coefficient of thermal expansion usually increases at elevated temperature in FGMs. However, material density still remains constant, even when the temperature increases. To predict the behaviour of FGMs under high temperature more accurately, it is necessary to consider the temperature dependent on material properties. The nonlinear equation of thermo-elastic material properties as a function of temperature ܶሺܭሻ can be expressed as (Touloukian, 1967)

ିଵ ଶ ଷ ܲൌܲ଴ሺܲିଵܶ ൅ͳ൅ܲଵܶ൅ܲଶܶ ൅ܲଷܶ ሻǡ (4.4)

where ܶൌܶ଴ ൅ȟܶ and ܶ଴ ൌ ͵ͲͲܭ (ambient or free stress temperature), ȟܶ is the temperature change; ܲ଴ǡܲିଵǡܲଵǡܲଶ and ܲଷ are the temperature dependent coefficients which can be seen in the table of material properties (Table 4.1) for various types of materials.

Chapter 4

Table 4.1 Temperature dependent coefficients of Young’s modulus ܧ(Pa), thermal expansion coefficient ߙ (1/K), Poisson’s ratio ߥ, and mass density ߩ (kg/m3) for various materials (Shen, 2009)

Materials P0 P-1 P1 P2 P3 P, @ 300 K Al2O3 E 349.55e+9 0 -3.853e-4 4.027e-7 -1.673e-10 320.24e+9 ߙ 6.8269e-6 0 1.838e-4 0 0 7.203e-6 ߥ 0.26 0 0 0 0 0.260 ߩ 3800 0 0 0 0 3800 Si3N4 E 348.43e+9 0 -3.070e-4 2.160e-7 -8.946e-11 322.27e+9 ߙ 5.8723e-6 0 9.095e-4 0 0 7.475e-6 ߥ 0.24 0 0 0 0 0.240 ߩ 2370 0 0 0 0 2370 ZrO2 E 244.27e+9 0 -1.371e-3 1.214e-6 -3.681e-10 168.06e+9 ߙ 12.766e-6 0 -1.491e-3 1.006e-5 -6.778e-11 18.591e-6 ߥ 0.2882 0 1.133e-4 0 0 0.298 ߩ 3657 0 0 0 0 3657 SUS304 E 201.04e+9 0 3.079e-4 -6.534e-7 0 207.79e+9 ߙ 12.330e-6 0 8.086e-4 0 0 15.321e-6 ߥ 0.3262 0 -2.002e-4 3.797e-7 0 0.318 ߩ 8166 0 0 0 0 8166

4.1.2 Thermal buckling of FG beams based on the improved TSDT

Thermal buckling response of FG beams is not available in the open literature; hence, the validation study for the theoretical formulation in this research investigation is achieved by comparing the results with a symmetric three-layer (0o/90o/0o) cross-ply beam studied by Aydogdu and Lee (Aydogdu, 2007, Lee, 1997). The material properties of the laminated composite beam are temperature independent which are given as:

ܧଵΤܧଶ ൌʹͲ, ܩଵଶ ൌܩଵଷ ൌͲǤ͸ܧଶ, ܩଶଷ ൌͲǤͷܧଶ, ߥଵଶ ൌͲǤʹͷ, ߙଶΤߙଵ ൌ͵

ଶ ଶ and the dimensionless thermal buckling in this instance is ߣൌȟܶ௖௥ܮ ߙଵȀ݄ . Similar to Ayadogdu and Lee (Aydogdu, 2007, Lee, 1997), the thermal moment resultant and higher order terms in the potential energy equation of Eq. (3.19) are neglected. Table 4.2 shows dimensionless thermal buckling of the cross-ply beams with L/h=10 derived from various beam theories for different boundary conditions. It is observed that there is a good 99

Chapter 4

agreement of results with the published ones, especially for parabolic shear deformation beam theory (PSDBT).

ଶ ଶ Table 4.2 Dimensionless thermal buckling ሺߣൌȟܶ௖௥ܮ ߙଵȀ݄ ሻ of symmetric cross-ply beams (0o/90o/0o) with L/h=10

Source H-H H-C C-C Present 0.791 1.230 1.800 PSDBT [a] 0.790 1.230 1.797 HOBT [b] 0.823 1.280 1.871 FSDT [b] 0.828 1.290 1.886 [a] (Aydogdu, 2007) [b] (Lee, 1997)

ଶ ଶ Dimensionless thermal buckling for FG beams can be defined as ߣൌȟܶ௖௥ܮ ߙ௠Ȁ݄ in which ߙ௠is the coefficient of thermal expansion of SUS304 at ambient temperature.

For FG beam analysis in this research, various types of ceramics such as Alumina (Al2O3),

Silicon nitride (Si3N4) and Zirconia (ZrO2), which are widely used to make FGMs, are taken into consideration by mixing with Stainless steel (SUS304). The material properties of those materials are presented in Table 4.1. Table 4.3 presents dimensionless thermal buckling of FG beams with different end conditions H-H, H-C and C-C, and convergence studies for the Ritz method are adopted to achieve accurate solution. It is noted that increasing the number of polynomial terms leads to greater accuracy. The FG beams used in these studies are made of Si3N4/SUS304 with the material volume fraction index (n=0.5) and the length-to height or slenderness ratio (L/h=15).

100

Chapter 4

Table 4.3 Convergence studies for thermal buckling of Si3N4/SUS304 beams of temperature independent with n=0.5 and L/h=15

Գ H-H H-C C-C 2 1.168 1.988 3.874 4 0.963 1.922 3.657 6 0.962 1.922 3.654 7 0.962 1.922 3.655 8 0.962 1.922 3.655

Consequently, eight polynomial terms are large enough for every boundary condition to define the solution of thermal buckling of FG beams. Գൌͺ is, therefore, the upper limit number of the polynomial trial function in Eq. (3.28) that will be used to calculate in all of the thermal buckling analyses.

The thermal buckling results of various types of ceramics mixed with SUS304 are presented in Table 4.4. The hinged-hinged (H-H) FG beams produced from different pairs of materials with L/h=20 are investigated by varying the number of the volume fraction index (n). In this research, the Solution I refers to the solution derived from temperature independent material properties, while the Solution II is denoted as temperature dependent material property solution. As expected, the Solution I always provides higher values of thermal buckling results than those derived from the Solution II. It can be seen that the thermal buckling results decrease when the values of the power law index (n) increases, except for the pair of ZrO2/SUS304. In the case of a pure ceramic beam such as ZrO2 beam, this has a lesser value of thermal buckling result than that of the pure metal beam of

SUS304. It is confirmed that ZrO2 is not a suitable material in thermal barrier to mix with SUS304.

101

Chapter 4

ଶ ଶ Table 4.4 Dimensionless thermal buckling ሺߣൌȟܶ௖௥ܮ ߙ௠Ȁ݄ ሻ of H-H beams with L/h=20

Material Solution Solution Material Solution Solution Material Solution Solution I II I II I II Si3N4 1.348 1.185 Al2O3 1.376 1.326 ZrO2 0.518 0.416 n=0.2 1.103 0.991 n=0.2 1.124 1.066 n=0.2 0.534 0.437 n=0.5 0.968 0.882 n=0.5 0.979 0.923 n=0.5 0.551 0.459 n=1.0 0.876 0.805 n=1.0 0.880 0.827 n=1.0 0.565 0.481 n=2.0 0.810 0.749 n=2.0 0.810 0.759 n=2.0 0.576 0.503 n=5.0 0.750 0.697 n=5.0 0.747 0.698 n=5.0 0.589 0.530 n=10.0 0.712 0.664 n=10.0 0.708 0.662 n=10.0 0.600 0.549 SUS304 0.618 0.582 SUS304 0.618 0.582 SUS304 0.618 0.582

The relationship between dimensionless thermal buckling (ߣ) and slenderness ratio (L/h) is plotted in Fig. 4.3. In this figure, the clamped-clamped (C-C) FG beams made of

Si3N4/SUS304 with n=0.3 are taken into consideration by using both the solution types. With the increase in slenderness ratio, this leads to the slight changes in the thermal buckling results based on the Solution I, whereas, the considerable changes happen with the results derived from the Solution II. It is noted that the thermal buckling results obtained from the Solution II play a significant role in comparison with those obtained from the Solution I, especially for the cases of the thick FG beams that have low values of slenderness ratio. The small graph inside the Fig. 4.3 shows the amount of iteration before receiving the convergent results of the Solution II only. There is more initial fluctuation of the thermal buckling result for the thick FG beams than that of thin FG beams, which usually can obtain the convergent results with a small number of iterations.

102

Chapter 4

ଶ ଶ Fig. 4.3 Dimensionless thermal buckling ሺߣൌȟܶ௖௥ܮ ߙ௠Ȁ݄ ሻ of C-C beams made of Si3N4/SUS304 with n=0.3

Dimensionless thermal buckling results of different boundary conditions are considered in

Fig. 4.4 by varying the slenderness ratio. The FG beams made of Al2O3/SUS304 with n=0.3 are chosen to investigate in this analysis. The thermal buckling results obtained from the Solution II still show the significant features to be considered for all types of boundary conditions, particularly for small values of the slenderness ratio. However, the discrepancy between the Solution I and Solution II lowers considerably with the increase in the slenderness ratio. As expected, the thermal buckling results of C-C FG beams present the greatest values, followed with H-C and H-H end conditions, respectively.

103

Chapter 4

ଶ ଶ Fig. 4.4 Dimensionless thermal buckling ሺߣൌȟܶ௖௥ܮ ߙ௠Ȁ݄ ሻ of Al2O3/SUS304 beams with n=0.3

The critical buckling temperature of C-C FG beam made of Al2O3/SUS304 is presented in Fig. 4.5. Both solution types are used in this investigation with different slenderness ratio and the volume fraction index (n) of the FG beams. The dramatic changes in the values of the critical buckling temperature can be seen at the region of n=0.0-1.0, which is the region of the significant reduction of the percentage of Al2O3, and then continuing with the slight changes after that region for every slenderness ratio. The considerable effects of temperature-dependent on the material properties are observed for the beams with small slenderness ratio.

104

Chapter 4

Fig. 4.5 Critical buckling temperature (K) of C-C beams made of Al2O3/SUS304

The critical buckling temperature resulting from different pairs of materials to make FG beams as seen with Si3N4/SUS304 and ZrO2/SUS304 are taken into consideration with various values of the volume fraction index (n) as illustrated in Fig. 4.6. The FG beams in this investigation are H-C FG beams with L/h=30. It is observed that the critical buckling temperature is reduced when the volume fraction index (n) increases for the Si3N4/SUS304 beams. In contrast, for the case of ZrO2/SUS304 beams, increasing n leads to obtaining higher values of the critical buckling temperature.

105

Chapter 4

Fig. 4.6 Critical buckling temperature (K) of H-C beams with L/h=30

4.1.3 Thermo-elastic vibration of FG beams based on the improved TSDT

From the extensive literature review on this research topic, it is clear that there are no results available on the free vibration analysis of the heated FG beams. Hence, the theoretical formulation in this research is verified by setting ∆T=0 in Eq. (3.19), and then comparing free vibration results of FG beams without thermal effects of Sina et al. (Sina et al., 2009) and Simsek (Şimşek, 2010a). Sina et al. (Sina et al., 2009) show more accuracy can be obtained from the developed first order theory FSDT1 than that of the traditional first order theory FSDT2 in order to predict frequency results of FG beams. In their research, the FG beams made of Al/Al2O3 where the Al rich is set at the top surface, while the opposite surface is the Al2O3 rich, whose material properties are

3 Al: ܧ௠ ൌ͹Ͳ GPa, ߩ௠ ൌ ʹ͹ͲͲ kg/m , ߥ௠ ൌͲǤʹ͵,

3 Al2O3: ܧ௖ ൌ ͵ͺͲ GPa, ߩ௠ ൌ ͵ͺͲͲ kg/m , ߥ௠ ൌͲǤʹ͵.

106

Chapter 4

Dimensionless frequency ሺȳሻ for the FG beam can be defined as

ܫ଴ ȳൌ߱ܮଶ Ǥ ௛ (4.5) ඩ ଶ ଶ ሺݖሻ݀ݖܧ ׬ି௛ ݄ ଶ

The displacement and rotation field indices required in the Ritz method for different boundary conditions used in Table 4.5 can be seen in Table 3.1. It is observed that the agreement is satisfactory in comparison between the present frequency results and those obtained from the published results for various boundary conditions and slenderness ratio, especially comparing with the developed theory FSDT1. The deviation in the results can be attributed to the higher order and improved terms of the new TSDT used in this research.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of Al/Al2O3 ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮTable 4.5 Dimensionless fundamental frequency ቆȳ ൌ ߱ beams with n=0.3 under ambient temperature

B.C. source L/h=10 L/h=30 L/h=100 Present 2.803 2.845 2.850 S-S FSDT1[a] 2.774 2.813 2.817 FSDT2[a] 2.695 2.737 2.742 PSDBT [b] 2.702 2.738 2.742 ASDBT [b] 2.702 2.738 2.742

Present 1.008 1.015 1.016 FSDT1[a] 0.996 1.003 1.003 C-F FSDT2[a] 0.969 0.976 0.977 PSDBT [b] 0.970 0.976 0.977 ASDBT [b] 0.970 0.976 0.977

Present 6.078 6.416 6.459 FSDT1[a] 6.013 6.343 6.384 C-C FSDT2[a] 5.811 6.167 6.212 PSDBT [b] 5.881 6.177 6.214 ASDBT [b] 5.884 6.177 6.214 [a] (Sina et al., 2009) and [b] (Şimşek, 2010a)

Dimensionless fundamental frequencies of H-H FG beams made of Al2O3/SUS304 with

L/h=30 based on the Solution I in relation to the range of high percentage of Al2O3 (n=0.0- 1.0) are presented in Fig. 4.7. The beam ends are hinged end hence the beam is presented from extending as the temperature is increased. According to the study of Li et al. (Li et al., 107

Chapter 4

2004), the fundamental frequency of isotropic beams decreases with the increase of temperature until it approaches the critical buckling temperature. The fundamental frequency tends to zero at the critical temperature before increasing at the post-buckling region. Similarly, for FGMs in the study of Abrate (Abrate, 2008), the author concluded that FGMs behave like isotropic homogeneous materials in the framework of linear analysis. However, for the case of nonlinear analysis, this may show different behaviour between both kinds of materials due to the effect of coupling between in-plane and bending deformation. As a result, it is seen that Al2O3/SUS304 beam with n=0.2, for example, the fundamental frequency declines to zero at its critical temperature (οܶ௖௥ ൌ ͺͳǤͻͳܭ derived from the section of thermal buckling analysis) after that temperature the frequency increases.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of H-H beams ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.7 Dimensionless fundamental frequency ቆȳ ൌ ߱

made of Al2O3/SUS304 with L/h=30 derived from the solution I

108

Chapter 4

The thermal effects on material properties for dimensionless fundamental frequencies of FG beams are presented in Fig. 4.8 in order to compare the frequency results obtained from the Solution I and Solution II. The C-C FG beams made of Al2O3/SUS304 with L/h=30 are taken into consideration in this figure. It can be seen that the deviation between the Solution I and Solution II is very small at the low temperature changes. At the range of temperature before the critical temperature, it is found that the frequency results of the Solution I are higher than those of the Solution II for both values of the power law index (n=0.2 and n=2.0). However, this behaviour is opposite in the range of temperature beyond the critical temperature. Moreover, the FG beams with a higher percentage of Al2O3 (n=0.2) usually provide larger values of the frequency results.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of C-C beams ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.8 Dimensionless fundamental frequency ቆȳ ൌ ߱

made of Al2O3/SUS304 with L/h=30

109

Chapter 4

Dimensionless fundamental frequencies obtained from the Solution II for different boundary conditions with the variation of temperature changes are plotted in Fig. 4.9. The

FG beams made of Si3N4/SUS304 with L/h=30 and n=0.5 are chosen in this investigation. The boundary conditions used in this figure as seen with H-H, H-C and C-C whose critical temperature changes are 67.46, 131.16 and 238.15 K respectively. Therefore, the frequency result in each type of boundary condition declines to zero at its critical temperature before going up after that temperature.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of different ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.9 Dimensionless fundamental frequency ቆȳ ൌ ߱

boundary conditions for the FG beams made of Si3N4/SUS304 with L/h=30 and n=0.5

Dimensionless fundamental frequency is shown in Fig. 4.10 for the H-C FG beam made of

Si3N4/SUS304 for different slenderness ratio and the variation of the power law index (n). The frequency results in this figure are evaluated by the Solution II with the temperature change (οܶ) at 100 K. For the L/h=30 beam, it is observed that the fundamental frequency

110

Chapter 4

declines dramatically when the value of n increases until reaching the minimum point at n=8.5, following which it increases further. This behaviour can be explained that οܶ ൌ

ͳͲͲ K is the critical temperature of L/h=30 beam composed from Si3N4 and SUS304 with the power law index around n=8.5. While this temperature is much less than the critical temperature of L/h=20 and L/h=25 beams, it implies that the fundamental frequency of these beams will never go down to reach the minimum point with οܶ ൌ ͳͲͲ K.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of H-C beams ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.10 Dimensionless fundamental frequency ቆȳ ൌ ߱

made of Si3N4/SUS304 of the Solution II at ∆T=100 K

Dimensionless fundamental frequency of H-C FG beams composed from different pairs of materials is shown in Fig. 4.11. The slenderness ratio L/h=15 and the temperature change ∆T=150 K are selected for analysis using the Solution II. The fundamental frequency results decrease with the increment of the volume fraction index (n) for the pair of Si3N4/SUS304 and Al2O3/SUS304, but not for the ZrO2/SUS304 beam. There are slightly higher in value frequency results of Al2O3/SUS304 beam in comparison with those of Si3N4/SUS304 beam

111

Chapter 4

for every value of n. On the other hand, the ZrO2/SUS304 beam shows different behaviour and provides less value of the fundamental frequency results as well.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of H-C beams ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 4.11 Dimensionless fundamental frequency ቆȳ ൌ ߱ made of various materials of the Solution II at ∆T=150 K with L/h=15

4.2 Application of the improved TSDT to FG plate analysis

An FG plate made of ceramic-metal according to the power law distribution is considered in this section. The geometry of FG plate is shown in Fig. 4.12, in which the material composition at the top surface is assumed to be the ceramic-rich surface and the material compositions are varied continuously to the metal-rich surface at the opposite side.

112

Chapter 4

b a -h/2 z y

h/2 h Fig. 4.12 FG plate geometry

4.2.1 Thermal buckling of FG plates based on the improved TSDT

For the validation exercise for the thermal buckling analysis using the improved TSDT in this research, the thermal buckling results of simply supported FG plates presented in the published work of Javaheri et al. (Javaheri and Eslami, 2002b) are adopted to compare with the present results. Temperature independent material properties and the displacement and rotation functions corresponding to simply supported boundary conditions (SS-1) which are used in this calculation for the Solution I are given the same as Javaheri et al. (Javaheri and

Eslami, 2002b). The critical buckling temperature ΔTcr (K) is associated with uniform temperature rise of the FG plate made from the combination of material which consists of

Al2O3 and Al. Young’s modulus and coefficient of thermal expansion for Al2O3 are Ec=380 -6 0 -6 0 GPa, αc=7.4×10 / C and for Al are Em=70 GPa, αm=23×10 / C, respectively. Poisson’s ratio is given to be the same as 0.3. In Table 4.6, the numerical results of movable (SS-1) and immovable (SS-2) boundary conditions are presented and compared with the published results. It is observed that all of the buckling temperature results of SS-1 are lower than those of SS-2 for FG plates only. Due to the vanishing of the effect of bending-coupling stiffness B11 and warping-extensional coupling stiffness E11 for isotropic plates like a pure

Al2O3 and Al plate, this leads to obtaining the same results for SS-1 and SS-2. It is noted that all of the higher material stiffness components and the improved terms of the theory are being taken into account in this research, hence, this reveals the significant feature when the plate become thicker b/h≤20. 113

Chapter 4

Table 4.6 Comparisons of Critical buckling temperature (K) of Al2O3/Al plates based on the Solution I material Source b/h=10 b/h=20 b/h=40 b/h=60 b/h=80 Present (SS-2) 1595.148 419.856 106.386 47.402 26.687 Al2O3 Present (SS-1) 1595.148 419.856 106.386 47.402 26.687 [a] 1617.484 421.516 106.492 47.424 26.693

Present (SS-2) 862.044 226.464 57.353 25.552 14.385 n=1.0 Present (SS-1) 747.629 195.511 49.453 22.027 12.400 [a] 757.891 196.257 49.500 22.037 12.402

Present (SS-2) 809.660 217.448 55.405 24.713 13.982 n=5.0 Present (SS-1) 668.942 177.774 45.164 20.134 11.337 [a] 678.926 178.528 45.213 20.144 11.340 n=10.0 Present (SS-2) 761.634 204.929 52.243 23.304 13.126 Present (SS-1) 681.745 182.325 46.402 20.692 11.653 [a] 692.519 183.141 46.455 20.703 11.657 Al Present (SS-2) 513.222 135.084 34.228 15.251 8.586 Present (SS-1) 513.222 135.084 34.228 15.251 8.586 [a] (Javaheri and Eslami, 2002b)

Fig. 4.13 shows the critical buckling temperature (K) of FG plate made of Si3N4/SUS304 derived from temperature independent material properties (the Solution I) and the temperature dependent material properties (the Solution II). The results of temperature dependent material properties (the Solution II) play the significant role to be used in this problem, in particular, when the plate becomes thicker. It is seen that the thickness ratio

(h/b) of the Si3N4/SUS304 plate with n=0.5 could be limited at around 0.1 for the case of the simply supported (SS-1) boundary condition; and for the case of clamped (CC) boundary condition, it would be less than 0.1 to be considered by thermal buckling analysis. The plates which are thicker than the limitations could be evaluated by failure analysis due to thermal stresses, because their critical buckling temperatures are too high and close to the melting point temperature of the materials.

114

Chapter 4

1200 (a) 900 (b) 800 1000 Solution I Solution I 700 Solution II Solution II 800 600

cr 500 cr T

600 T Δ Δ 400

400 300

200 200 100

0 0 0.02 0.04 0.06 0.08 0.1 0.02 0.03 0.04 0.05 h/b h/b

Fig. 4.13 Critical buckling temperature (K) of Si3N4/SUS304 square plates with n=0.5: (a) SS-1 (b) CC

The critical buckling temperatures of Al2O3/SUS304 plates with various values of the volume fraction index (n) for immovable type of simply supported plates are presented in Table 4.7. The buckling temperature results in this table are derived from the Solution I and Solution II for various values of the thickness ratio (b/h).

115

Chapter 4

Table 4.7 Critical buckling temperature (K) of Al2O3/SUS304 plates (b/a=1.0)

Material Source b/h=15 b/h=20 b/h=30 b/h=50 b/h=80

Al2O3 Solution I 781.390 445.344 199.826 72.293 28.287 Solution II 696.822 415.290 193.323 71.394 28.149 n=0.2 Solution I 637.901 363.653 163.201 59.048 23.106 Solution II 544.430 329.608 155.725 58.005 22.943 n=1.0 Solution I 499.958 285.039 127.928 46.288 18.113 Solution II 418.505 254.167 120.841 45.286 17.955 n=2.0 Solution I 459.696 262.198 117.714 42.599 16.670 Solution II 384.028 233.102 110.932 41.631 16.517 n=5.0 Solution I 423.153 241.465 108.443 39.251 15.361 Solution II 353.356 214.423 102.097 38.340 15.217 SUS304 Solution I 350.531 199.947 89.771 32.488 12.713 Solution II 294.242 179.119 85.075 31.815 12.609

To investigate the thermal buckling analysis of a variety of materials, three different types of ceramics are chosen to mix with stainless steel (SUS304) in Table 4.8 for investigating thermal buckling analysis. To ensure a realistic situation that the buckling temperature results could be lower than the melting point of any material, the value of h/b=0.025 is chosen to calculate with various aspect ratio (b/a). The critical buckling temperature of FG plates with SS-2 is carried out in Table 4.8 using different values of the aspect ratio (b/a), height to side or thickness ratio (h/b), material composition and the volume fraction index (n). The results presented in Table 4.8 cover both the solution types. The parametric study in Table 4.8 reveals that the critical temperature is directly proportional to the thickness ratio and to the relative aspect ratio for all pairs of materials.

116

Chapter 4

Table 4.8 Critical buckling temperature (K) of FG plates made of different pairs of materials

Material Solution b/a=1 b/a=2 b/a=3 Material Solution b/a=1 b/a=2 b/a=3 Material Solution b/a=1 b/a=2 b/a=3

Al2O3 (I) 112.781 280.132 554.320 Si3N4 (I) 110.451 274.381 543.054 ZrO2 (I) 42.412 105.318 208.311 (II) 110.608 267.678 509.137 (II) 102.923 234.957 418.053 (II) 37.633 81.139 134.866 n=0.5 (I) 80.286 199.440 394.690 n=0.5 (I) 79.392 197.235 390.377 n=0.5 (I) 45.169 112.155 221.800 (II) 77.813 185.412 346.089 (II) 75.490 176.026 320.199 (II) 40.980 90.065 152.082 n=1.0 (I) 72.208 179.388 355.032 n=1.0 (I) 71.845 178.501 353.324 n=1.0 (I) 46.281 114.927 227.316 (II) 69.831 166.074 309.656 (II) 68.644 160.990 295.042 (II) 42.504 94.610 161.968 n=5.0 (I) 61.224 152.013 300.588 n=5.0 (I) 61.536 152.799 302.178 n=5.0 (I) 48.277 119.898 237.202 (II) 59.074 140.184 261.181 (II) 59.169 139.811 258.951 (II) 45.689 105.453 188.244 SUS304 (I) 50.677 125.823 248.807 SUS304 (I) 50.677 125.823 248.807 SUS304 (I) 50.677 125.823 248.807 (II) 49.110 116.971 217.849 (II) 49.110 116.971 217.849 (II) 49.110 116.971 217.849

117

Chapter 4

As presented above, the Solution II is an appropriate solution for the real life applications, and it also shows the significant feature for analysing thermal buckling of FG plates; therefore, it is used to obtain the critical buckling temperature for the different pairs of materials in Fig. 4.14. The buckling temperature in this figure is obtained from TSDT with an immovable boundary condition (SS-2), and the volume fraction index (n) is fixed as n=0.5. It is seen that Al2O3/SUS304 plate provides the largest values of buckling temperature and is followed with the results of Si3N4/SUS304 and ZrO2/SUS304 plates, respectively.

Al2O3/SUS304 Si3N4/SUS304 ZrO2/SUS304 1000 900 800 700 600 cr

T 500 Δ 400 300 200 100 0 123456789101112131415 b/a

Fig. 4.14 Critical buckling temperature (K) of FG plates made out of different pairs of materials (n=0.5, h/b=0.01)

The critical buckling temperature of immovable boundary condition (SS-2) obtained from the Solution II is presented in Fig. 4.15. The FG plates made of three different pairs of materials are used to investigate their buckling temperature by varying the power low index (n). Increasing the value of n leads to decreasing the value of buckling temperature of

Al2O3/SUS304 and Si3N4/SUS304 plates; while, ZrO2/SUS304 plate shows the opposite behaviour.

118

Chapter 4

Al2O3/SUS304 Si3N4/SUS304 ZrO2/SUS304 800

700

600

500 cr

T 400 Δ

300

200

100

0 0246810 n Fig. 4.15 Critical buckling temperature (K) of FG plates made out of different pairs of materials (a/b=1.0, h/b=0.08)

To investigate the thermal buckling analysis of a fully clamped (CC) FG plate, the displacement and rotation functions of Eq. (3.73) are used to substitute into the total energy functional of Eq. (3.67). Consequently, the critical buckling temperatures of CC plates made of Al2O3/SUS304 with several values of the volume fraction index are presented in Table 4.9. The buckling temperature results in this table are derived from both the Solution I and Solution II with different thickness ratios.

119

Chapter 4

Table 4.9 Critical buckling temperature (K) of fully clamped plates made of Al2O3/SUS304 (b/a=1.0)

Material Source b/h=15 b/h=20 b/h=30 b/h=50 b/h=80

Al2O3 Solution I 1990.426 1170.786 548.973 219.171 103.634 Solution II 1564.252 997.459 504.618 211.387 101.829 n=0.2 Solution I 1619.402 953.012 446.906 178.273 84.132 Solution II 1163.199 764.355 397.159 169.339 82.040 n=1.0 Solution I 1255.029 738.585 346.280 138.023 65.035 Solution II 867.946 576.716 302.174 129.869 63.107 n=2.0 Solution I 1149.535 676.884 317.186 125.998 58.952 Solution II 806.491 529.826 276.140 118.332 57.157 n=5.0 Solution I 1061.942 625.766 293.108 116.030 53.888 Solution II 769.226 493.011 255.030 108.861 52.228 SUS304 Solution I 888.903 523.564 245.364 97.434 45.548 Solution II 630.301 412.728 215.167 91.933 44.273

To consider the influence of materials on the critical buckling temperatures of the fully clamped FG plates for which their results are presented in Table 4.10, three types of ceramics, Al2O3, Si3N4 and ZrO2 are chosen to mix with SUS304. Based on the numerical results, increasing the values of n yield the reduction in the critical buckling temperatures of the Al2O3/SUS304 and Si3N4/SUS304 plates, except for ZrO2/SUS304 plates in which the trend is reversed. Similarly with the case of thermal buckling analysis of FG beams, it is also found that the pure ZrO2 plates give lower critical temperatures than those of the pure

SUS304 plates for every aspect ratio. Therefore, it is reasonable to conclude that ZrO2 is the only type of ceramic, which is not recommended to mix with SUS304 to make FG plates owing to its inferior material property.

120

Chapter 4

Table 4.10 Critical buckling temperature (K) of fully clamped FG plates made of different pairs of materials (h/b=0.025)

Material Solution b/a=0.5 b/a=1.0 b/a=2.0 Material Solution b/a=0.5 b/a=1.0 b/a=2.0 Material Solution b/a=0.5 b/a=1.0 b/a=2.0

Al2O3 (I) 246.735 324.839 882.684 Si3N4 (I) 242.400 318.954 865.709 ZrO2 (I) 92.247 121.554 330.992 (II) 236.939 308.281 777.428 (II) 210.691 267.729 604.501 (II) 73.006 90.788 183.682 n=0.5 (I) 174.205 229.362 623.290 n=0.5 (I) 172.537 227.105 616.787 n=0.5 (I) 97.752 128.861 351.236 (II) 163.332 211.133 513.561 (II) 155.919 199.796 466.131 (II) 80.255 100.494 208.052 n=1.0 (I) 155.483 204.762 556.924 n=1.0 (I) 154.899 203.958 554.415 n=1.0 (I) 100.219 132.077 359.791 (II) 145.267 187.733 456.149 (II) 141.433 181.715 429.390 (II) 84.169 105.853 223.055 n=5.0 (I) 130.977 172.814 471.971 n=5.0 (I) 131.789 173.857 474.609 n=5.0 (I) 105.150 138.486 376.671 (II) 121.997 157.882 387.287 (II) 121.923 157.491 382.237 (II) 93.851 119.677 269.498 SUS304 (I) 109.885 144.860 394.877 SUS304 (I) 109.885 144.860 394.877 SUS304 (I) 109.885 144.860 394.877 (II) 102.983 133.298 325.444 (II) 102.983 133.298 325.444 (II) 102.983 133.298 325.444

121

Chapter 4

1200 Al2O3/SUS304 900 (a) Al2O3/SUS304 (b) 1000 Si3N4/SUS304 800 Si3N4/SUS304 ZrO2/SUS304 700 ZrO2/SUS304 800 600

cr 500 cr

T 600 T Δ Δ 400 400 300

200 200 100

0 0 1357 0510 b/a n Fig. 4.16 Critical buckling temperature (K) of fully clamped FG plates made of different pairs of materials; (a) effect of plate aspect ratio and (b) effect of the volume fraction index

As shown in Fig. 4.16, the fully clamped FG plates are made from different pairs of materials which are Al2O3/SUS304, Si3N4/SUS304 and ZrO2/SUS304, in order to investigate the effects of the plate aspect ratio and volume fraction index on the critical buckling temperature of the fully clamped FG plates. The results are calculated using the Solution II with h/b=0.01, n=0.5 for (a) and b/a=1, h/b=0.05 for (b). It is observed that the variations of the buckling temperature results present a similar trend of changes compared to the previous case of simply supported boundary condition when the considered effects are varied. In general, it is also found that the buckling temperature results of fully clamped FG plates are much higher than those of simply supported FG plates, for every pair of materials.

4.2.2 Thermo-elastic vibration of FG plates based on the improved TSDT

ଵȀଶ Dimensionless frequencies ൫ȳ ൌ ݄߱ሺߩ௧Ȁܧ௧ሻ ൯ of FG plates without considering thermal effect are calculated with the omission of the potential energy equation due to thermal stresses in Eq. 3.60. As a case study, the first three modes of the dimensionless frequencies 122

Chapter 4

of Al2O3/Al plates under ambient temperature are presented in Table 4.11. The material properties considered in this table are taken from published literature (Hosseini-Hashemi et al., 2011), which have E=70 GPa, ρ=2702 kg/m3, ν=0.3 for Al and E=380 GPa, ρ=3800 3 kg/m , ν=0.3 for Al2O3. It can be observed that there is a very good agreement between the present results and those found in the open literature, especially when compared with a new exact solution of Hosseini-Hashemi (Hosseini-Hashemi et al., 2011) and the numerical HSDT solution of Matsunaga, (Matsunaga, 2008) according to movable simply supported (SS-1) boundary condition.

ଵȀଶ Table 4.11 Dimensionless frequencies ൫ȳ ൌ ݄߱ሺߩ௧Ȁܧ௧ሻ ൯ of Al2O3/Al plates under ambient temperature (a/b=1.0)

h/a=0.1 h/a=0.2 mode Source n=0.5 n=1.0 n=10.0 n=0.5 n=1.0 n=10.0 (1,1) Present 0.0490 0.0442 0.0364 0.1807 0.1631 0.1301 Exact [a] 0.0490 0.0442 0.0366 0.1805 0.1631 0.1324 HSDT [b] 0.0492 0.0443 0.0364 0.1819 0.1640 0.1306 FSDT [c] 0.0482 0.0435 0.0359 0.1757 0.1587 0.1284 (1,2) Present 0.1174 0.1059 0.0856 0.3989 0.3607 0.2771 Exact [a] 0.1173 0.1059 0.0867 0.3978 0.3604 0.2856 HSDT [b] 0.1180 0.1063 0.0859 0.4040 0.3644 0.2790 FSDT [c] 0.1154 0.1042 0.0850 - - - (2,2) Present 0.1807 0.1631 0.1301 0.5803 0.5254 0.3948 Exact [a] 0.1805 0.1631 0.1324 0.5779 0.5245 0.4097 HSDT [b] 0.1819 0.1640 0.1306 0.5891 0.5444 0.3981 FSDT [c] 0.1764 0.1594 0.1289 - - - [a] (Hosseini-Hashemi et al., 2011) ,[b] (Matsunaga, 2008), [c] (Zhao et al., 2009a)

For different pairs of materials, Table 4.12 provides the first three modes of dimensionless frequencies of FG plates. The frequency results are calculated from the improved TSDT under ambient temperature. The material properties in each type of material at 300 K temperature are given in Table 4.1. A close agreement of the results compared with the published one for the case of Si3N4/SUS304 plate can be observed in this table. Other plates made from Al2O3/SUS304 and ZrO2/SUS304 are also taken into consideration.

123

Chapter 4

The frequency results are presented in the form of dimensionless frequencies as

ଶ ଶ ଵȀଶ ȳൌሺ߱ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ where ܧ଴ and ߩ଴ are the reference values of ܧ௕ and ߩ௕ at T0=300 K.

ଶ ଶ ଵȀଶ Table 4.12 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of FG plates under ambient temperature (a=b=0.2, h=0.025)

n=0 n=0.5 n=1.0 n=2.0

Mode Si3N4/SUS304 present [a] present [a] present [a] present [a] (1,1) 12.507 12.495 8.646 8.675 7.599 7.555 6.825 6.777 (1,2) 29.256 29.131 20.080 20.262 17.705 17.649 15.947 15.809 (2,2) 44.323 43.845 29.908 30.359 26.727 26.606 24.147 23.806

Al2O3/SUS304 (1,1) 9.841 - 7.803 - 7.114 - 6.563 - (1,2) 23.008 - 18.253 - 16.633 - 15.323 - (2,2) 34.794 - 27.569 - 24.700 - 23.048 -

ZrO2/SUS304 (1,1) 7.260 - 6.368 - 6.037 - 5.753 - (1,2) 16.953 - 14.824 - 14.014 - 13.294 - (2,2) 25.671 - 24.570 - 21.456 - 20.247 - [a] (Huang and Shen, 2004)

The dimensionless fundamental frequencies of Si3N4/SUS304 plate with different temperature rises are presented in Table 4.13. Both types of solutions, the Solution I and II, are used to calculate the frequency results for SS-1 and SS-2 boundary conditions. The frequency results decrease as the increase in volume fraction index (n), for every value of temperature rise.

124

Chapter 4

ଶ ଶ ଵȀଶ Table 4.13 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of

Si3N4/SUS304 plates (a/b=1.0; h/b=0.1)

οܶ B.C. Source Material Si3N4 n=0.2 n=0.5 n=1.0 n=2.0 n=5.0 SUS304 SS-1 Solution I 11.958 9.529 7.991 6.923 6.154 5.538 4.720 200 Solution II 11.610 9.228 7.722 6.677 5.925 5.324 4.512 SS-2 Solution I 11.958 9.542 8.030 6.983 6.214 5.570 4.720 Solution II 11.610 9.241 7.761 6.737 5.986 5.356 4.512 SS-1 Solution I 11.110 8.665 7.133 6.070 5.309 4.697 3.782 400 Solution II 10.218 7.810 6.307 5.269 4.540 3.964 2.966 SS-2 Solution I 11.110 8.679 7.176 6.137 5.377 4.734 3.782 Solution II 10.218 7.828 6.364 5.361 4.637 4.017 2.966 SS-1 Solution I 10.192 7.705 6.157 5.076 4.300 3.668 2.517 600 Solution II 8.423 5.863 4.263 3.136 2.378 1.805 1.474 SS-2 Solution I 10.192 7.721 6.206 5.155 4.384 3.715 2.517 Solution II 8.423 5.897 4.380 3.353 2.640 1.978 1.474

Dimensionless fundamental frequencies of Si3N4/SUS304 square plates with the volume fraction index n=0.5 and 5.0 according to the Solution I and II are presented in Fig. 4.17 by varying the values of temperature change. It is seen that the values of dimensionless frequencies decrease to their minimum point close to zero at the critical buckling temperature before increasing after that temperature. For example, in the case of the FG plate with n=0.5 the critical buckling temperature which can be derived from the section of thermal buckling analysis, is ΔTcr=690.650 and 511.642 K of the Solution I and II, respectively. At the pre-buckling region, the frequency results of the Solution I are higher than those of the Solution II; however, this trend is opposite for the post-buckling region.

125

Chapter 4

10 n=0.5 Solution I Solution II 9 8 7 6

Ω 5 4 n=5.0 3 2 1 0 0 200 400 600 800 ΔT

ଶ ଶ ଵȀଶ Fig. 4.17 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of Si3N4/SUS304 plates (a/b=1.0; h/b=0.075; SS-2)

Dimensionless fundamental frequencies of Si3N4/SUS304 square plates are plotted in three- dimensional form as shown in Fig. 4.18. All the frequency results considered in this figure are in the pre-buckling region. It is noted that the frequency results decrease as the increment of the volume fraction index (n) and the temperature change (ΔT).

126

Chapter 4

8 7 6 5 4Ω 3 2 1 00

200

1 400 ∆T 2 4 n 6 8 600 10

ଶ ଶ ଵȀଶ Fig. 4.18 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of Si3N4/SUS304 plates (a/b=1.0; h/b=0.10; SS-2; the Solution II)

The dimensionless fundamental frequencies of Si3N4/SUS304 square plates using the Solution II with the immovable boundary condition (SS-2) and various values of the volume fraction index (n) under the temperature difference ΔT=200 K are shown in Fig. 4.19. For the FG plate having the thickness ratio b/h=20, its frequency results decrease to the minimum point at the volume fraction index (n) around n=13.3 before increasing again.

This means that ΔT=200 K is the critical buckling temperature for the Si3N4/SUS304 plates with b/h=20 and n=13.3. However, for the thicker plates of b/h=10 and 15, their frequency results will never decline to reach the minimum point close to zero with ΔT=200 K. This is because their critical buckling temperatures are much higher than ΔT=200 K.

127 Chapter 4

10 b/h=10 b/h=15 b/h=20 9

Ω 5

0 0 2 4 6 8 10 12 14 n

ଶ ଶ ଵȀଶ Fig. 4.19 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of Si3N4/SUS304 plates

Fig. 4.20 presents dimensionless fundamental frequencies of FG square plates made from three different pairs of materials. The frequency results in this figure are obtained from the Solution II of the immovable boundary condition (SS-2).

128

Chapter 4

10 Si3N4/SUS304 Al2O3/SUS304 ZrO2/SUS304 9

Ω 5

0 0 100 200 300 400 500 600 700 800 ΔT

ଶ ଶ ଵȀଶ Fig. 4.20 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of FG plates (a/b=1.0; h/b=0.075; n=0.5)

In order to investigate thermo-elastic vibration of a fully clamped FG plate, the displacement and rotation functions of Eq. (3.69) are multiplied by ݁௜ఠ௧ as the harmonic vibration and then substituted into the total energy functional of Eq. (3.64). Consequently, the natural frequencies of the clamped FG plates can be obtained. The numerical results of fundamental frequencies of clamped Si3N4/SUS304 plates with various values of the volume fraction index are tabulated in Table 4.14 by varying the temperature change parameters from ΔT=0 to ΔT=500 K. Two types of solutions that are the Solution I and II are used to find out the frequency results in this table.

129

Chapter 4

ଶ ଶ ଵȀଶ Table 4.14 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of fully

clamped FG plates made of Si3N4/SUS304 (a/b=1.0; h/a=0.1)

Material ΔT=0 ΔT=100 ΔT=200 ΔT=300 ΔT=400 ΔT=500

Si3N4 Solution I 23.869 23.610 23.348 23.083 22.815 22.544 Solution II 23.869 23.349 22.820 22.274 21.703 21.098 n=0.2 Solution I 19.240 18.982 18.719 18.453 18.183 17.908 Solution II 19.240 18.777 18.279 17.742 17.158 16.522 n=0.5 Solution I 16.358 16.107 15.851 15.591 15.327 15.057 Solution II 16.358 15.938 15.467 14.939 14.351 13.696 n=1.0 Solution I 14.303 14.058 13.807 13.552 13.292 13.026 Solution II 14.303 13.915 13.462 12.941 12.348 11.679 n=2.0 Solution I 12.760 12.519 12.274 12.023 11.766 11.504 Solution II 12.760 12.397 11.959 11.445 10.852 10.179 n=5.0 Solution I 11.518 11.282 11.040 10.792 10.538 10.278 Solution II 11.518 11.175 10.749 10.242 9.653 8.983 SUS304 Solution I 10.146 9.897 9.641 9.377 9.106 8.825 Solution II 10.146 9.809 9.372 8.836 8.203 7.472

A parametric study is carried out by varying temperature and plate aspect ratio of fully clamped Si3N4/SUS304 plates for fundamental frequencies, using both solution types. The numerical results of the dimensionless fundamental frequencies of such plates are presented in Table 4.15. It is evident that increasing the aspect ratio leads to reduction in frequency results for all values of ΔT.

130

Chapter 4

ଶ ଶ ଵȀଶ Table 4.15 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of fully

clamped FG plates made of Si3N4/SUS304 (h/a=0.1; n=0.5)

b/a ΔT=0 ΔT=100 ΔT=200 ΔT=300 ΔT=400 ΔT=500 0.5 Solution I 37.331 37.061 36.788 36.513 36.236 35.956 Solution II 37.331 36.694 35.983 35.190 34.307 33.321 1.0 Solution I 16.358 16.107 15.851 15.591 15.327 15.057 Solution II 16.358 15.938 15.468 14.939 14.351 13.696 1.5 Solution I 12.812 12.578 12.340 12.097 11.849 11.595 Solution II 12.812 12.444 12.030 11.566 11.048 10.470 2.0 Solution I 11.722 11.500 11.273 11.041 10.804 10.561 Solution II 11.722 11.376 10.987 10.550 10.060 9.514

To study the influence of the volume fraction index on frequency results of fully clamped

Si3N4/SUS304 plates, the fundamental frequencies of such plates are presented in Table 4.16 with various values of the volume fraction index (n) and the thickness ratio (a/h). In this table, the numerical results written in the parentheses refer to the frequencies in unit of rad/s. As expected, the frequencies decrease when the plates become thinner with the increase of the thickness ratio. Moreover, the frequencies are reduced by increasing values of n for every aspect ratio.

131

Chapter 4

ଶ ଶ ଵȀଶ Table 4.16 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of fully

clamped FG plates made of Si3N4/SUS304 (a/b=1) a/h Source n=0 n=0.2 n=0.5 n=1.0 n=2.0 n=5.0 n=10.0 10 Solution I 22.563 18.183 15.327 13.292 11.766 10.538 10.002 (1185.583) (955.421) (805.349) (698.427) (618.262) (553.729) (525.574) Solution II 21.447 17.158 14.351 12.348 10.852 9.653 9.124 (1126.962) (901.581) (754.071) (648.849) (570.238) (507.226) (479.429) 15 Solution I 25.135 20.143 16.865 14.499 12.698 11.243 10.624 (586.998) (470.419) (393.846) (338.602) (296.540) (262.570) (248.117) Solution II 23.601 18.733 15.522 13.201 11.437 10.017 9.407 (551.174) (437.470) (362.497 (308.291) (267.092) (233.934) (219.678) 20 Solution I 27.674 22.054 18.332 15.604 13.484 11.759 11.051 (363.535) (289.711) (240.812) (204.983) (177.125) (154.464) (145.173) Solution II 25.647 20.185 16.550 13.873 11.785 10.086 9.384 (336.913) (265.157) (217.403) (182.242) (154.817) (132.488) (123.268) ( ) Frequency results in unit of rad/s

In Table 4.17, the fundamental frequency results of fully clamped FG plates made from three different pairs of materials are presented with various values of the volume fraction index and temperature change, using the Solution I and II. It is found that the ZrO2/SUS304 plates provide lower frequency results than those of other plates.

ଶ ଶ ଵȀଶ Table 4.17 Dimensionless frequencies ൫ȳ ൌ ሺ߱ ܽ Τ݄ሻሾߩ଴ሺͳ െ ߥ ሻȀܧ଴ሿ ൯ of fully clamped FG plates made of different pairs of materials (a/b=1; h/a=0.1)

Si3N4/SUS304 Al2O3/SUS304 ZrO2/SUS304 n ΔT=100 ΔT=200 ΔT=300 ΔT=100 ΔT=200 ΔT=300 ΔT=100 ΔT=200 ΔT=300 0.5 I 16.107 15.851 15.591 14.517 14.289 14.057 11.633 11.292 10.940 II 15.938 15.468 14.939 14.384 14.003 13.592 11.160 10.156 8.802 1.0 I 14.058 13.807 13.552 13.144 12.911 12.673 11.049 10.735 10.410 II 13.915 13.462 12.941 13.025 12.631 12.188 10.683 9.843 8.739 5.0 I 11.282 11.040 10.792 11.065 10.826 10.581 10.173 9.899 9.618 II 11.175 10.749 10.242 10.965 10.556 10.073 9.980 9.409 8.702

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4.2.3 Thermo-elastic forced vibration of FG plates based on the improved TSDT

This section is given to investigate and present forced vibration response of FG plates supported by simply supported and fully clamped boundary conditions. The FG plates are subjected to the uniformly distributed dynamic load acting on the plate domain under thermal environment. Beginning with the simply supported FG plates, the maximum dimensionless deflections of Si3N4/SUS304 plates due to the dynamic load are tabulated in Table 4.18. To verify accuracy, the available deflection results of Alibeigloo and Kari (Alibeigloo and Kari, 2009) are adopted to compare with the present results, for the case of isotropic plates only by setting n=0. The new deflection results of FG plates are also presented with various values of volume fraction index (n). It is defined that D0=D11 is for an isotropic plate made of the same material at the bottom surface of the FG plate.

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ of isotropic and FGܦTable 4.18 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ plates (a/b=1.0; a/h=10.0; SS-2)

Isotropic plates FG plates ݎఠ [a] n=0 n=0.5 n=1.0 n=5.0 n=10.0 0.0 0.427 0.425 0.325 0.343 0.380 0.396 0.1 0.432 0.429 0.328 0.347 0.384 0.400 0.3 0.471 0.468 0.358 0.378 0.418 0.436 0.5 0.574 0.570 0.436 0.461 0.510 0.532 0.7 0.849 0.845 0.646 0.683 0.756 0.788 0.9 2.301 2.290 1.751 1.852 2.049 2.136 [a] (Alibeigloo and Kari, 2009)

To investigate on the thermal effect, the maximum deflection results measured at the plate centre of the FG plates are presented in Table 4.19, using ΔT=300 K. Both solution types are employed to compute the deflection results. It is clearly seen that the deflection results obtained from the Solution I are much lower than those of the other solution type.

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ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ of Si3N4/SUS304 platesܦTable 4.19 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ (a/b=1.0; a/h=10.0; ΔT=300 K; SS-2)

n=0.5 n=1.0 n=5.0

ݎఠ Solution I Solution II Solution I Solution II Solution I Solution II 0 0.436 0.500 0.479 0.556 0.568 0.675 0.1 0.441 0.505 0.484 0.562 0.574 0.682 0.3 0.480 0.551 0.527 0.612 0.625 0.744 0.5 0.585 0.671 0.642 0.746 0.761 0.905 0.7 0.865 0.992 0.950 1.102 1.126 1.338 0.9 2.341 2.682 2.569 2.980 3.044 3.616

The deflection results of Al2O3/SUS304 plates are plotted against frequency ratio which is the proportion of external frequency to natural frequency of the plate system, as shown in Fig. 4.21. The Solution II is adopted to obtain the deflection results in relation to different temperature changes. The considerable changes of deflections are observed at the resonance zone in which the external frequency corresponds to the natural frequency of the system. As expected, the deflection results increase as the temperatures increase.

3 ΔT=100 K ΔT=200 K ΔT=300 K 2.5

ݓഥ 1.5

0.5

0 0 0.2 0.4 0.6 0.8 1 ݎఠ

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ of Al2O3/SUS304 platesܦFig. 4.21 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ (n=0.5, a/b=1.0; a/h=10.0; SS-2; Solution II)

134

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In Fig. 4.22, the relationship between deflection and frequency ratio which is associated with different values of volume fraction index (n) are shown. The deflection changes follow the same trend as the previous figure, in which the deflections change slightly at the zone far from the resonance point (ݎఠ ൌͳǤͲ) and they approach infinity at the resonance point. Additionally, it is found that increasing the values of n leads to increase of deflections, for

Al2O3/SUS304 plates.

4 n=0.5 n=1.0 n=5.0 3.5

2.5

ݓഥ 2

1.5

0.5

0 0 0.2 0.4 0.6 0.8 1 ݎఠ

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ of Al2O3/SUS304 platesܦFig. 4.22 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ (a/b=1.0; a/h=10.0; SS-2; ΔT=300 K; Solution II )

Fig. 4.23 illustrates the deflection results due to forced vibration of FG plates made from three different pairs of materials. It is clearly seen that the deflection result of

ZrO2/SUS304 plate is much greater than that of Al2O3/SUS304 and Si3N4/SUS304 plates which show very similar results with each other, for every value of the frequency ratio.

135

Chapter 4

12 Al2O3/SUS304

10 ZrO2/SUS304 Si3N4/SUS304 8 ݓഥ 6

0 0 0.2 0.4 0.6 0.8 1 ݎఠ

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ of FG plates made ofܦFig. 4.23 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ different pairs of materials (a/b=1.0; a/h=10.0; SS-2; ΔT=300 K; Solution II )

To consider forced vibration analysis of fully clamped FG plates subjected to the uniformly distributed dynamic loads, the assumed displacement and rotation functions as seen in Eq. 3.69 that can satisfy the fully clamped conditions are substituted into the total energy functional in Eq. 3.66. The maximum deflection results of the fully clamped plates made of

Al2O3/SUS304 are shown in Fig. 4.24 with the variation of temperature values. When compared to the case of simply supported FG plates, the trend of deflection changes corresponding to the variations of temperatures and frequency ratios is similar which shows the increasing trend as those parameters increase. However, the deflection values detected in the case of the fully clamped plates are much lower than those of simply supported plates. This is because the fully clamped plates are much stiffer than simply supported plates.

136

Chapter 4

1.6 ΔT=100 K ΔT=200 K ΔT=300 K 1.4

1.2

ݓഥ 0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 ݎ ఠ

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ of Al2O3/SUS304 platesܦFig. 4.24 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ (n=0.5, a/b=1.0; a/h=10.0; CC; Solution II)

In order to consider the influence of the volume fraction index on deflection results, Fig.

4.25 gives the maximum deflection results of Al2O3/SUS304 plates with different values of the volume fraction index based on the Solution II. The increasing temperature parameter is fixed as ΔT=300 K. As illustrated in the figure, it is seen that increasing the volume fraction index yields the increase in deflection of the Al2O3/SUS304 plate for every frequency ratio.

137

Chapter 4

1.8 n=0.5 n=1.0 n=5.0 1.6

1.4

1.2

ݓഥ 1

0.8

0.6

0.4

0.2

0 0 0.2 0.4 0.6 0.8 1 ݎఠ

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ of Al2O3/SUS304 platesܦFig. 4.25 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ (a/b=1.0; a/h=10.0; CC; ΔT=300 K; Solution II )

Table 4.20 presents the numerical results of dimensionless deflections of fully clamped FG plates made of three different pairs of materials. A slight difference between the deflection results of Al2O3/SUS304 plates and Si3N4/SUS304 plates is observed when the volume fraction index (n) and the frequency ratio are varied. For these plates, it is also observed that increasing the n value leads to an increase in the deflection result. In contrast, when comparing the deflection results of these two plates with those of ZrO2/SUS304 plates, it is found that the deflection results of ZrO2/SUS304 plates are much greater. Another difference is that increasing the value of n for the case of ZrO2/SUS304 plates leads to reduction in the deflection result.

138

Chapter 4

ସ ଴Τሻܲ଴ܽ ሻݓ௠௔௫ of fully clamped FGܦTable 4.20 Dimensionless deflections ሺݓഥ ൌ ሺͳͲͲ plates made of different pairs of materials (a/b=1.0; a/h=10.0; ΔT=300 K; Solution II)

Al2O3/SUS304 Si3N4/SUS304 ZrO2/SUS304 ݎఠ n=0.5 n=1.0 n=5.0 n=0.5 n=1.0 n=5.0 n=0.5 n=1.0 n=5.0 0 0.137 0.150 0.174 0.140 0.152 0.175 0.345 0.303 0.242 0.1 0.139 0.151 0.176 0.142 0.154 0.177 0.349 0.307 0.244 0.3 0.153 0.167 0.193 0.156 0.170 0.195 0.386 0.339 0.270 0.5 0.191 0.209 0.241 0.196 0.213 0.243 0.489 0.429 0.340 0.6 0.231 0.252 0.290 0.237 0.257 0.293 0.598 0.523 0.414 0.7 0.307 0.335 0.381 0.315 0.342 0.386 0.811 0.708 0.557 0.8 0.491 0.534 0.598 0.506 0.548 0.606 1.376 1.187 0.923 0.9 1.507 1.631 1.650 1.598 1.715 1.701 6.412 5.054 3.542

139

Chapter 5 FGM fabrication

In this chapter the description of the procedure to manufacture the functionally graded (FG) materials is presented. In this research, FGM specimens were manufactured from the mixture of ceramics and polymers. Alumina (Al2O3) is used for the ceramic constituent. Epoxy-resin is used for the polymer constituent. Based on the extensive literature review in Chapter 2, it was found that, among three effective techniques used for manufacturing FGMs; namely, thermal spraying, powder metallurgy and infiltration techniques, the ceramic-polymer FGMs can only be manufactured appropriately by using the infiltration technique. Therefore, a multi-step sequential infiltration technique is used to fabricate FG beam specimens made from alumina and epoxy-resin in this research. In this chapter, the information in relation to FGM fabrication will be provided in detail as follows:

5.1 Specimen fabrication using multi-step sequential infiltration technique 5.2 Microstructure analysis 5.3 Alternative technique of specimen fabrication

5.1 Specimen Fabrication using multi-step sequential infiltration technique

Alumina-epoxy composite beams were manufactured using a multi-step sequential infiltration technique. The earliest development of the technique was proposed by Cichocki et al. (Cichocki et al., 1998) to produce graded composite materials.

140

Chapter 5

The key principle of the infiltration technique is to make an alumina piece containing a graded network of porosity which will be infiltrated by epoxy in the final step, in order to obtain alumina-epoxy composite specimens. By using this technique, the composite specimens can be represented in the form of interpenetrating-network (IPN) structured composites which contain two phases of materials that are alumina and epoxy phases. In several previous investigations (Clark, 1992, Prielipp et al., 1995, Lange et al., 1990), it was affirmed that the infiltration technique is an effective methodology that can be used for producing graded composite materials where their material compositions are continuous. Composite materials having the interpenetrating network structures are different from traditional fibre-matrix and particle-matrix composite materials in terms of compositional distribution and pattern which then lead to different behaviour when they are subjected to mechanical loads. Therefore, this chapter will give overviews of the production process for making graded composite beams.

The unique advantage of the infiltration technique is that it is suitable for making graded composites with a wide range of material constituents. As mentioned in Chapter 2 of the literature review, for example, Neubrand et al. (Neubrand et al., 2002) successfully employed the infiltration technique to produce ceramic-metal graded composites which were used for investigating residual stress analysis. To obtain graded composites which have a large disparity in elastic properties, ceramic-polymer could be used as the material constituents. Thus, in the investigations of Tilbrook et al. (Tilbrook et al., 2006), the alumina-epoxy graded composite system in which material compositions were varied along an axial direction, was produced using the infiltration technique for the purpose of investigating crack propagation. In this research, the material compositions between alumina and epoxy of the FG beam specimens will be graded across the thickness direction of the specimens. The outline of the fabricating process is illustrated in Fig. 5.1.

141

Chapter 5

Compress

Polyurethane Al2O3 slip casting foam →drying

180 C Al 2

O 3

(a) Plaster mount (b)

Burning out Infiltrating foam & Sintering Epoxy Resin ceramic

Epoxy

Silicon mount (c) (d)

Alumina-rich

Epoxy-Resin rich (e)

Fig. 5.1 Schematic of fabricating process of graded composite specimen

142

Chapter 5

5.1.1 Foam preparation

According to the schematic of the production process, it began with the foam preparation process. This is to construct a porous network structure inside the specimens. The network structure was achieved by using open-celled polyurethane foam (PU) as an imprint for such a structure. The polyurethane foam used in this preparation was commercial foam whose average cell size was approximately 0.8 mm. The PU foam was cut to the required sizes (length 100 mm and breadth 12 mm), then the foam pieces were sliced to three different thicknesses (h), 2 mm, 4 mm and 6 mm. There were many ways to slice the foam pieces in order to obtain a series of very thin foams. For instance, in the foam preparation process of (Tilbrook, 2005), the PU foam was immersed with water and frozen to retain shape before it was cut to size with a bandsaw. This method of cutting the foam did not yield a perfect result because the cut foam pieces were wet and required thawing and drying later. Usually, the drying process of the wet foam pieces took around 12 hrs to dry in an oven at 45 0C. Therefore, in this fabrication, the PU foam was cut and sliced by using a hot-wire cutter that can be considered as a better way, without requiring any process of freezing, thawing and drying.

The foam pieces with different thicknesses were placed in a uniaxial hot press and compressed to the same constant thickness of 1 mm, producing a series of foam pieces of different density. In order to retain the compressed shape, a heat treatment cycle had to be applied during this compressing of foam. A schematic of the foam compression process is shown in Fig. 5.2. As seen in Fig. 5.2, a pressure of 300 kPa was applied to compress the foam pieces. The temperature was ramped up to 180 0C with the increasing rate of 1 0C/min, and was held for 1 hour before letting the temperature cool down naturally to room temperature. The foam was then removed from the hot press. Unevenness at the edges due to thermal stress was found which was removed before being utilised in the subsequent stages.

143

Chapter 5

300 kPa

Heated 180 0C for 1 hr

Fig. 5.2 Schematic of foam preparation for the porous network imprint

The uniaxial hot press that was used in this foam preparation process is shown in Fig. 5.3. It is seen that the brass spacer containing the foam pieces having different thicknesses was positioned between the top and bottom platforms. Both platforms were set to heat up to 180 0C using a temperature control box. The pressure of the hot press was controlled at 300 kPa by an hydraulic system. The spacer under the mentioned pressure and temperature was held for 1 hour in order to ensure that the foam pieces inside had retained the new shapes. The spacer was then allowed to cool down naturally before removal of the compressed foam pieces.

144

Chapter 5

Top platform

Brass spacer Temperature control box

Bottom platform

Fig. 5.3 The uniaxial hot press used to produce a series of different density foam pieces

5.1.2 Slip casting and drying processes

To manufacture graded composite specimens, a slip casting process was used to create alumina phases in specimens. A useful and appropriate optimum slip formula for the infiltration technique was determined by Neubrand et al. (Neubrand et al., 1999). Therefore, the alumina slip used to manufacture the specimens in this research was produced using the same formula as proposed in (Neubrand et al., 1999, Tilbrook, 2005). Within the slip formula, several additives were incorporated in order to improve the quality of the slip casting process with a colloidal suspension of very fine alumina particles. For example, a wetting agent was used to aid foam infiltration, an antifoaming agent to prevent air entrapment, and binders and plasticisers to create green strength and prevent drying cracks (Rutgers, 2005). The slip contents were presented as follows:

145

Chapter 5

Alumina Slip Formula

45 mL of alumina slip with a 25vol% solids loading can be obtained with the following formula:

x 32.525 g Distilled water x 0.145 g Ammonia solution (25% concentrate) x 0.464 g Dolapix CE64 (Zschimmer and Schwarz, Lahnstein, Germany) x 0.276 g Glydol N 109 NEU (Zschimmer and Schwarz, Lahnstein, Germany) x 0.094 g Contraspum KWE (Zschimmer and Schwarz, Lahnstein, Germany) x 0.927 g Duramax B-1052 (Rohm & Haas, Lauterborg, France) x 46.350 g Alumina Powder (Taimicron TM-DAR, Taimei Chemical Co Ltd, Japan)

The 99.99 % purity alumina powder (Al2O3), powder size of approximately 1μm, was chosen to make the alumina slip because small powder particles can penetrate easily into the pores of the compressed foam. All of the chemical ingredients used to make the alumina slip in this research are shown in Fig. 5.4.

146

Chapter 5

Glydol Contraspum Duramax

Weighing machine

Alumina powder

Ammonia Alumina slip Dolapix

Fig. 5.4 Chemical ingredients used to produce alumina slip for slip casting process and the amount of the ingredients measured by using the 4 digit weighing machine

A very good mixture of all ingredients (the liquid components) with the alumina powder according to the formula can be achieved by the aid of a magnetic stirrer (IEC magnetic stirrer) and an ultrasonic bath (UNISONICS). All of the chemical ingredients and alumina powder were weighed sequentially corresponding to the formula and mixed together in a beaker. A teflon-coated magnetic bar was added to the beaker, which was then placed on the magnetic stirrer. The magnetic stirrer was operated for 10 minutes before the beaker was moved to the ultrasonic bath for 10 minutes. However, one cycle of mixing, was found to be insufficient to obtain a good mixture. Therefore, the mixing process with the magnetic stirrer and the ultrasonic bath was repeated again. The mixing process was carried out at least 3 times in order to produce the strongest mixture of all ingredients. It was found that the mixing process led to a reduction of large agglomerates and produced good dispersion. The alumina slip was prepared one day before casting for ageing purposes. Before the alumina slip was used for casting in the following day, it was re-mixed or re-dispersed by the magnetic stirrer and the ultrasonic bath again.

147

Chapter 5

In the slip casting process, the casting mould, which consists of pieces of Perspex, was used to create the graded composite specimens. The Perspex was made from plaster of Paris. The pieces of the Perspex were fitted together to the dimension of the compressed foam (length 100 mm and breadth 12 mm). They were placed on a Plaster of Paris block in a vacuum chamber, as shown in Fig. 5.5. The benefit of the Plaster of Paris block is that it can absorb moisture from the alumina slip and let the specimen dry appropriately. To minimise the problem of adhesion between the surfaces of the cast specimen to the mould walls, the Perspex pieces were lined with Teflon tape. The filter paper (ADVANTEC) was placed between the compressed foam and the plaster of Paris block. This is done in order to avoid the likelihood of contamination of the alumina slip by particles or soluble ions from the plaster. To clearly understand the slip casting set up, the cross-section of apparatus is illustrated in Fig. 5.5.

Perspex wall Teflon tape Specimen Filter paper

Plaster of Paris Vacuum chamber frame Fig. 5.5 Schematic cross-section of slip casting experimental set up

After preparing the casting mould, the series of compressed foam pieces were stacked into the mould. The highest density foam piece was placed in the mould first, followed by the lighter ones. Due to the density of the foam pieces being much lighter than that of the alumina slip, the foam pieces would normally float in the slip causing the problem of discontinuity between adjacent layers. To solve this problem, there were several attempts to glue the foam pieces together in order to ensure phase connectivity across the step- interfaces. Using glue to stick the foam pieces together was one of many attempts to create the connectivity; however, the glue tended to cause warping of the foam, producing delamination. Another attempt was to use a heat treatment with slight pressure to construct the foam piece connection. However, these foam joining techniques presented above were not effective in producing a suitable bond. Thus a better way to join the foam pieces was to 148

Chapter 5

place the foam pieces so they were restrained slightly by the mould walls or the Perspex pieces without using glue and then apply a little pressure when placing them into the mould. Consequently, good connectivity between the adjacent layers can be obtained.

Slip casting was performed in the vacuum chamber (LABEC) by placing the mould assembly into the chamber. At the beginning stage of the slip casting process, the alumina slip was filled into the foam pieces at a slow filling rate and using a small amount of the slip; otherwise, the foam pieces would have floated on the surface. To enhance infiltration of the slip casting process after the slip was poured over the foam pieces the pressure of the chamber was lowered to around 100 mbar that led to air bubbles leaving the foam. When the lowered pressure was released to atmospheric pressure, the slip was then forced into the porosity of the foam pieces. To minimise the amount of air bubbles, the cycle of varying pressure could be repeated several times.

After leaving the casting specimen within the mould for 1 or 2 days, the water was observed to have been absorbed by the Plaster of Paris block. As a result, the slip volume decreased. At this stage, the foam pieces were joined and glued together by more colloid alumina slip. To complete the slip casting process, an additional slip was required with the cycle of varying pressure again. The mould containing the specimen was kept in a closed plastic container to prevent rapid moisture evaporation from the top surface, which might result in drying cracks. After being kept in the plastic container for around 7 days, the specimen had hardened sufficiently to retain the shape and was then removed from the casting mould. The set of the casting mould and some sufficiently dry specimens are shown in Fig. 5.6.

149

Chapter 5

(a) (b)

Fig. 5.6 (a) The set of casting mould consisting of Perspex and Plaster of Paris block; (b) examples of sufficiently dry specimens

The dry specimen was relocated onto a piece of uncompressed polyurethane foam in order to aid air movement to all surface areas in another closed container. The specimens were kept in the container for around 7 days. Subsequently, the lid of the container was gradually opened to promote slow moisture removal from the specimen within the period of 3-5 days. The final drying process was to move the specimen into an environmental chamber at 50 0C for 12 hours.

5.1.3 Foam burning out and sintering

After the casting specimen dried completely, the graded porous network structures inside the specimen were then manufactured by burning out the compressed foam pieces. The specimen whose size was 100 mm x 12 mm x 4 mm (length x breadth x thickness) was fired with a low temperature regime, to first pyrolyse the foam pieces and debinder the ceramic phase. The temperature profile used for the whole process of foam pyrolysis is illustrated in Fig. 5.7 with the details of temperature increment. For example, starting with the room temperature, the following temperature was used: 25 0C to 235 0C at 300C/hour held for 1 hour (foam burnout), 235 0C to 400 0C at 300C/hour held for 1 hour (binder

150

Chapter 5

burnout), 400 0C to 800 0C at 400C/hour held for 1 hour (ash burnout). The specimen was allowed to cool down naturally to room temperature before it was heated up again for sintering the ceramic phase at 1400 0C, held for 1 hour. The temperature profile of sintering process is shown in Fig. 5.8. The furnace, (Ceramic engineering furnace manufacturer, Sydney, Australia) as shown in Fig. 5.9, was used for both burning out foam and sintering ceramic processes. It was found that sintering reduces the dimensions of the specimen in every direction, and leaves an alumina specimen with layers of graduated porosities.

900 800 0C for 1 hr 800 ash burnout

700 Natural C)

0 600 cool 400 0C for 1 hr 400C/h ramp 500 binder burnout 400 235 0C for 1 hr 300 0 Temperature ( Temperature foam burnout 30 C/h ramp 200

100 300C/h ramp 0 0 5 10 15 20 25 30 Time (hours)

Fig. 5.7 Temperature profile for foam, binder and ash burn out

151

Chapter 5

1600 1400 0C for 1hr sintering 1400

1200 Natural cool C) 0 1000

800

600 100 0C/h ramp

Temperature ( Temperature 400

200

0 0 5 10 15 20 Time (hours)

Fig. 5.8 Temperature profile for sintering of the alumina phase

Fig. 5.9 Furnace used for burning out foam and sintering ceramic phase

152

Chapter 5

5.1.4 Polymer infiltration

The alumina specimen having layers of graduated porosities was ready for polymer infiltration, using a vacuum chamber, in order to obtain an alumina-epoxy composite specimen. The alumina specimen was placed into a silicone mould which was suitable for this process because the specimen can be removed easily when polymer is cured. Epoxy resin (Epofix, Struers, Denmark) was chosen to fill the graduated porosities inside the alumina specimen under varying pressures and was then cured at room temperature. By varying the pressure, air bubbles were forced to leave the specimen and epoxy resin could infiltrate the graduated porosities more easily. The cycles of varying pressure within a range of 100 to 1000 mbar were required until ensuring the amount of air bubbles was minimised. The next step was to leave the epoxy resin cured at room temperature for around 24 hours. After curing, excess epoxy was removed by polishing with coarse sandpaper. Finally, the surfaces of specimen were ground and polished with diamond paste.

5.2 Microstructure Analysis

The graded composite beam specimens made from alumina-epoxy which were produced using the above procedure were selected randomly to section across the thickness direction to examine the quality of the graded area. Then the sectioned specimen was mounted by epoxy resin before grinding and polishing with sandpapers and diamond paste, respectively. For polishing with the diamond paste, the specimen was first polished using an automatic polisher (Leco VP-50). The used diamond paste has a diamond particle size of 3 μm which was reduced to 1 μm at the final step.

Images of the polished surfaces, obtained by optical microscopy with a Nikon 200 microscope and digital camera, were used to characterise the pure alumina and epoxy phases over the graded region. One such cross-section image was shown in Fig. 5.10. A measurement of areas, using the areal technique, in the cross-section yielded the percentages of material compositions in each graded region. The top and bottom surfaces were referred to as the pure alumina and epoxy resin layers, respectively.

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Ceramic

(a) (b) (c)

(a) (b)

Fig. 5.10 Material compositions across the thickness direction of the specimen made of alumina and epoxy resin

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Fig. 5.11 illustrates the whole material composition at the graded region of the specimen. This includes information about percentages of alumina and epoxy resin in each layer across the graded region.

Fig. 5.11 Image of material compositions of alumina and epoxy resin at the whole graded region

As shown in Fig. 5.11: the cross-section image, a few pores were found due to imperfect epoxy infiltration. It was also seen that the variation of material compositions throughout the cross-section presented in the form of layer or step changes rather than a continuous change over the section. By using the areal technique, one can obtain the percentages of alumina and epoxy resin in each layer in the graded region. It is clearly seen that the top layer was for a pure alumina and followed with (70% alumina-30% epoxy), (40% alumina- 60% epoxy) and (20% alumina-80% epoxy) layers, whereas, the bottom layer was for a pure epoxy resin layer. According to the material compositions or material volume fraction throughout the cross-section, these were plotted and illustrated in Fig. 5.12. The area under

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the step graph as shown in Fig. 5.12 matches the percentage of alumina (Vc); the proportion of epoxy resin is indicated by the complementary area of the graph.

Layered FGM 1 Layered FGM 0.8

0.6

Vc 0.4

0.2

0 -0.5 -0.3 -0.1 0.1 0.3 0.5 z/h

Fig. 5.12 Alumina volume fraction Vc )( of the specimen

The process of slip casting was further investigated. A microstructure analysis displayed the result of an imperfectly graded composite specimen when the alumina slip was filled into the compressed foam pieces with a high filling rate and too much slip. The microstructure of the imperfect specimen that was made by a high rate of alumina slip infiltration is shown in Fig. 5.13. It can be seen that there are a number of buffer layers of alumina across the specimen (seen in white colour layers). This is due to the compressed foam’s density being much lighter than that of the slip. Therefore, the foam pieces easily float on the slip between layers so that the buffer layers are constructed. The buffer layers cause the problem of porosities occurring inside the specimen, especially at the middle zone of the specimen. It is because the buffer layers have blocked or sealed a way for the epoxy resin to come into the lower zone. This can be rectified by filling the alumina slip into the foam pieces with a slow filling rate. The foam pieces were allowed to absorb all of the slip in the previous infiltration before more slip was added to complete the slip casting process.

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Buffer layer

Porosity

Fig. 5.13 Microstructure of imperfect specimen with buffer layers and porosities

5.3 Alternative technique of specimen fabrication

The principles and ideas of powder metallurgy and multi-step sequential infiltration techniques as presented in Chapter 2 are combined to propose an alternative technique to produce layered graded specimens. The main advantage of the combined technique is that it does not require the processes of slip casting and drying; therefore, time taken to manufacture specimens is greatly reduced. The combined technique utilises the fact that small particles of mothballs can be used to mix with the ceramic powder in each layer with different compositions in order to produce graded porosities. The mothballs are burnt out at low temperature during the sintering process; this leaves the ceramic specimen having graded porosities. After sintering, the ceramic specimen is solidified and ready for the process of polymer infiltration, consequently, the graded composite specimen made from ceramic and polymer can be obtained.

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This proposed idea of a combined technique can be proven by manufacturing specimens from the new process. Mothballs are selected to mix with ceramic powder as they can be burnt out completely at around 200 oC. The production process of the specimens using the combined technique began with grinding mothballs to small particle sizes whose diameter are around 0.5-1.0 mm. The 99.99% purity alumina powder (Taimicron TM-DAR, Taimei Chemical Co Ltd, Japan) was mixed with the ground mothballs in different bottles which were desired to have different material compositions. In this fabrication, the five layer composite specimens were required with the percentages of materials in each layer being (100 vol.% alumina-0 vol.% mothball); (20 vol.% alumaina-80 vol.% mothball); (40 vol.% alumina-60 vol.% mothball) and (70 vol.% alumina-30 vol.% mothball). However, the last layer contains pure epoxy which can be created in the final step of polymer infiltration. Hence, four bottles with different percentages of alumina powder and ground mothballs were obtained. The alumina powder and ground mothballs with CMC bonder powder were then blended for every bottle by using a ball milling machine (Star Machinery Pty Ltd) as shown in Fig. 5.14. The ball milling machine was set to run for 4-5 hours at 200 rpm in order to obtain a good mixture of the material compositions.

(a) (b)

Fig. 5.14 (a) Mixing balls and a bottle containing alumina powder, ground mothballs and CMC bonder powder; (b) the ball milling machine used for mixing materials

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After the materials in each bottle were mixed, they were brought to fill and form in a steel die. Fig. 5.15 (a) illustrates the steel die. Fig. 5.15 (b) displays the press machine (Enerpac 13 series), which was operated at 30 kN to compress materials in the steel die. For layered graded specimens, it was noted that the mixed powders were stacked sequentially layer by layer with a layer-wise compositional distribution. The first layer (at the bottom of the die) was given for pure alumina layer and followed with (20 vol.% alumaina-80 vol.% mothball); (40 vol.% alumina-60 vol.% mothball) and (70 vol.% alumina-30 vol.% mothball), respectively.

(a) (b)

Fig. 5.15 (a) Steel die; (b) press machine used to compact mixed powders

The green compacted specimen was removed from the die slowly with a small application of force through the compressing bar of the die in order to avoid cracks. The green specimen was then sintered using the high temperature furnace which is shown in Fig. 5.9. The increasing temperature profile of this sintering was used on the same principle as presented in Fig. 5.8. During the sintering, the mothballs were burnt out. After sintering, one can obtain the alumina specimen having graded porosities. To complete the fabrication process, the graded porosities inside the alumina specimen were infiltrated by epoxy-resin

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(Epofix, Struers, Denmark) with the aid of varying pressure in the vacuum chamber as shown in Fig. 5.16.

Fig. 5.16 Vacuum chamber used to speed up epoxy-resin infiltration

The layered graded specimen made from alumina and epoxy using the combined technique was finalised in a similar fashion to the infiltration technique by grinding and polishing the surfaces. The specimen was then sectioned to examine the quality of material gradients across the graded direction using optical microscopy. It was found that the specimen obtained via the combined technique had much more porosity than that produced by the multi-step sequential infiltration technique. To show the combined technique leading to porosities inside the specimen, Fig. 5.17 presents diagrams of layered graded specimen fabrication via this technique. The diagrams can be used to explain the mechanisms of graded construction in the specimen. The process of layer preparation is shown in Fig. 5.17 (a). The mixtures of ground mothballs and alumina powder are used to create layers according to the desired layer-wise compositional distribution. The sintering process leads to mothballs being burnt out and leaves the alumina specimen with layers of graded porosities. The subsequent step is to infiltrate epoxy into the graded porosities of the 160

Chapter 5

alumina specimen as shown in Fig. 5.17 (b). It can be seen that the epoxy profile 1 cannot pass into the following layer, but the epoxy profile 2 can penetrate through the porosity connection. However, very few of the porosity connections are created inside the specimen using the combined technique. Therefore, there is much porosity left without epoxy infiltration in the specimen as shown in Fig. 5.17 (b).

(a) Graded layer 1

Graded layer 2

Graded layer 3

Alumina layer

Epoxy profile 1 Epoxy profile 2 Epoxy layer (b)

Graded layer 1

Graded layer 2

Graded layer 3

Alumina layer

Mothball Epoxy Pore

Fig. 5.17 Diagrams of producing layered graded specimen made of alumina and epoxy with the combined technique; (a) layer preparation process; (b) epoxy infiltration process

Due to few porosity connections, the final specimen produced from alumina and epoxy using the combined technique may have a lot of porosities as seen in Fig. 5.18. The specimen in this figure contains 3 phases: epoxy, alumina and porosity phases. Only the top

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and bottom layers can be classified as the perfect layers which are the layers of pure epoxy and pure alumina respectively.

Epoxy layer

Graded layer 1

Graded layer 2

Graded layer 3

Alumina layer

alumina Epoxy Pore

Fig. 5.18 Layered FGM specimen made of alumina and epoxy with porosities

According to the diagrams of layered graded specimen fabrication, it can be concluded that the combined technique which is developed from the combination of ideas of powder metallurgy and infiltration techniques is not suitable for manufacturing the layered graded specimen. This is because the combined technique cannot produce the interpenetrating- network (IPN) structures which are very important to minimise the amount of porosity. Hence, to produce the layered graded specimen made of alumina and epoxy, the multi-step sequential infiltration technique as presented in Section 5.1 is strongly recommended to be used for producing such specimen. In Fig. 5.11, it is clearly seen that the interpenetrating- network (IPN) structures are created in the specimen produced by the multi-step sequential infiltration technique and the number of porosities is minimal. Thus, the layered graded specimens fabricated by the multi-step sequential infiltration technique are used for further investigation of vibration testing in this research.

162

Chapter 6 Results and discussion on the experimental and analytical predictions

The majority of research investigations on the mechanics of functionally graded materials (FGMs) have mainly been confined to theoretical analysis. FGMs are advanced materials which have spatially varying properties. In theoretical analysis, the material constituents of FGMs have often been assumed to vary smoothly and continuously along the graded direction. However, the various manufacturing techniques used to fabricate functionally graded (FG) components cannot easily produce them with properties varying smoothly according to a continuous mathematical function. Rather, existing techniques produce layered, functionally graded components.

Layered FG beam specimens fabricated by the multi-step sequential infiltration technique described in Chapter 5 were tested for a vibration characteristic with several types of end supports. The layered beams were made of alumina and epoxy. The experimental results corresponding to different boundary conditions were predicted by an analytical solution based on the improved TSDT. The power law distribution, which is usually used to describe material properties of common FGMs, can be applied to find out the properties of the layered FGMs as well. For the layered FGMs, the power law index or the volume fraction index (n) calculated approximately from a layered graded direction is required. However, the last section of this chapter is devoted to analysis of the vibration response of continuous FG beams made of general ceramics and polymers in which their material properties are assumed to vary smoothly and continuously throughout the thickness. 163

Chapter 6

6.1 Material properties of the layered FG beams

A simple mathematical modeling based on the power law distribution used for approximating the volume fraction index (n) of the layered FGMs is introduced in this research. The mathematical modeling can be expressed as follows.

ே σ௞ୀଵ ሺܸ௔ሻ (6.1) ݊ൌ ݖ ͳ σே ሺ ൅ ሻ ௞ୀଵ ݄ ʹ where N is the number of layers through the graded region. By using the value of n that is calculated by using Eq. (6.1) with the power law relations, one can obtain the volume fraction of materials across the beam specimens made of alumina ሺܸ௔ሻ and epoxy ሺܸ௘ሻ as follows.

(ݖ ͳ ௡ (6.2 ܸ ൌ൬ ൅ ൰ ௔ ݄ ʹ

ܸ௘ ൌͳെܸ௔ (6.3)

It is observed from the image of the cross section obtained by optical microscopy with a Nikon 200 microscope and digital camera (see Fig. 5.11), the material compositions inside the specimen present in the form of step changes rather than continuous changes across the sample thickness. However, to predict mechanical properties of the layered FG material, an approximated value of the volume fraction index (n) is needed to find out the properties across the specimen thickness. In this case, by using Eq. (6.1) with the percentages of ሺݖሻܧ Alumina in the sample, ݊ൌͳǤ͵ can be obtained to further calculate Young’s modulus and density ߩሺݖሻ. The volume fraction of alumina of the specimen and its prediction using Eq. (6.2) with ݊ൌͳǤ͵ are illustrated in Fig. 6.1.

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Layered FGM Prediction 1 0.9 0.8 0.7 0.6

Vc 0.5 0.4 0.3 0.2 0.1 0 -0.5 -0.3 -0.1 0.1 0.3 0.5 z/h

Fig. 6.1 The volume fraction of alumina ሺܸ௖ሻ of the specimen and its predicted material profile (h=3 mm; n=1.3)

ሺݖሻ and densityܧ Using a power law distribution, the approximation of Young’s modulus :ߩሺݖሻ of the layered FG beams varies across the thickness direction (z) as

(ݖ ͳ ௡ (6.4a ǡ ܧሻ ൬ ൅ ൰ ൅ ܧെ ܧሺݖሻ ൌ ሺܧ ௔ ௘ ݄ ʹ ௘

(ݖ ͳ ௡ (6.4b ߩሺݖሻ ൌ ሺߩ െߩ ሻ ൬ ൅ ൰ ൅ߩ Ǥ ௔ ௘ ݄ ʹ ௘

Poisson’s ratio ሺߥሻ is assumed to be constant with the average value. The geometry of a layered FG beam made of alumina-epoxy is shown in Fig. 6.2, and an example of material compositions throughout the FG beam thickness is illustrated in Fig. 6.3.

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z Alumina-rich

h/2 x

-h/2 Epoxy-rich b L Fig. 6.2 Geometry of a layered functionally graded beam

Alumina

Epoxy

Fig. 6.3 Material compositions across the layered FG beam thickness

6.2 Vibration experiment of layered FG beams

In order to validate the theoretical formulation, experimental tests on the free vibration response of the layered FG beams manufactured from the multi-step infiltration technique were performed. The natural frequencies of the beams were measured for several types of boundary conditions such as clamped-free (C-F), Simply supported-Simply supported (S- S), clamped-simply supported (C-S) and clamped-clamped (C-C). In the simply supported boundary condition, a knife-edge support was used in order to simulate the theoretical

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simply-supported condition. A simple measurement of the vibration associated with tapping a beam was set up as shown in Fig. 6.4 for the case of C-F beam. A Bruel & Kjaer (B&K) 4368 accelerometer was used to measure the signal. Pulse Labshop software was employed to analyse the signal from the accelerometer. A pre-trigger delay was set in the computer program in order to exclude some of the initial parts of the recording where the signal may have been lost.

Fig. 6.4 Experimental set up for vibration testing of the layered functionally graded beam

6.3 The effect of added mass on natural frequencies on the layered FG beams

For a particular case of the weight of sample being small compared to that of instrument, in vibration analysis, the mass of the instrument would be taken into account as the effect of added mass. In general, natural frequencies of the system decrease gradually as the increment of mass magnitude. To consider such effect onto vibration analysis of beam, the

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kinetic energy equation due to added mass on the basic of the improved TSDT can be expressed as follows:

௫೘ାௗȀଶ ߲ݑ ଶ ߲ݓ ଶ ݄ ߲ݑ ߲߶ ߲ݑ ߲ଶݓ ܾ ܶ ൌ නߚܯ൥ቆ൬ ଴൰ ൅൬ ଴൰ ቇ൅ ቆͷ ଴ ௫ െ ଴ ଴ቇ (௔ ʹ ௗ ߲ݐ ߲ݐ ͸ ߲ݐ ߲ݐ ߲ݐ ߲ݔ߲ݐ (6.5 ௫೘ିௗȀଶ ଶ ଶ ଶ ଶ ଶ ߲߶௫ ߲߶௫ ߲ ݓ଴ ߲ ݓ଴ ݄ ൅ ൭ʹͷ ൬ ൰ െͳͲ ൅ቆ ቇ ൱൩ ݀ݔ ͳͶͶ ߲ݐ ߲ݐ ߲ݔ߲ݐ ߲ݔ߲ݐ

2 where ܯௗ is the magnitude of the added mass per unit area (kg/m ), and β is the mass ratio between the added mass and the mass of the beam. Hence, the total kinetic energy (T) is

ܶ୲୭୲ୟ୪ ൌܶ௕ ൅ܶ௔Ǥ (6.6) where ܶ௕ is the kinetic energy equation of the beam which represents the same equation as presented in Eq. (3.26) and ܶ௔ is the kinetic energy equation of the added mass. The strain energy equationሺܷ௘ሻ, which is required to construct the total energy functionalሺȫሻ, can be obtained from Eq. (3.16).

The total energy functionalሺȫሻ presented in this research can be used to predict the fundamental frequency of the layered FG beams made of Al2O3/Epoxy. The material properties of Al2O3 in terms of Young’s modulus (E) and density (ρ) are assumed to be the same as Ref. (Sina et al., 2009), and E=2.6 GPa and ρ=1200 kg/m3 are used for Epoxy (Reid and Zhou, 2000). However, the value of Poisson’s ratio is set to be the same as ν=0.3 for both such materials. It is noted that the experimental sample specimen made of

Al2O3/Epoxy was small (around 7.0 g); therefore, the mass of the accelerometer and a part of its cable (3.5 g) could have a noticeable effect on frequency. As a result, the kinetic energy equation of the added mass in Eq. (6.5) should be included in calculations. Vibration testing for the beam samples, which had b=11 mm and h=3 mm, was performed with different end conditions. The effective length for the cases of S-S, C-S and C-C beam was set to L=65 mm, while the length L=75 mm was set for the C-F beam. The accelerometer was placed at the center for the S-S, C-S and C-C beams, and for C-F beam

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it was placed close to the tip of the beam (around ¾ of the beam length, measured from the fixed end).

Experimental and theoretical frequency results ሺ݂ሻ in the unit of Hz for the layered FG beams with various boundary conditions are compared in Table 6.1. A reasonable correlation is observed between the experimental results and those predicted by theoretical formulation, for every boundary condition. The discrepancies between theoretical and experimentally observed frequencies were around 10 per cent for every boundary condition. Imperfect epoxy infiltration leading to porosity (see Fig. 5.11), and the error due to approximation of the material volume fraction profile (see Fig. 6.1), are sufficient explanation for these discrepancies.

Table 6.1 Comparisons between theoretical and experimental fundamental frequency ሺ݂ൌ߱Τሻ ʹߨ of the layered FG beam

Source C-F S-S C-S C-C Theory 329.53 (Hz) 1393.74 (Hz) 2136.60 (Hz) 2923.61 (Hz) Experiment 303 (Hz) 1278 (Hz) 1980 (Hz) 2642 (Hz) Diff% 8.7 9.1 7.9 10.6

The following investigations study other types of layered FG beams. Two more types of layered FG beams are analysed, assuming material compositions as shown in Fig. 6.5. The 6 and 7 layer beams are designated FGM II and FGM III respectively, while FGM I beams are used as described earlier (see Fig. 6.1). The area under the step graph or smooth predicted graph matches the percentage of Al2O3; the proportion of epoxy is indicated by the complementary area. It is clearly visible in the composition charts that FGM II beams have a higher percentage of Al2O3 than FGM III beams. The details for each type of the layered beam are expressed in Table 6.2.

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Layered FGM Prediction Layered FGM Prediction 1 1

0.8 0.8

0.6 0.6

Vc Vc 0.4 0.4

0.2 0.2

0 0 -0.5 0 0.5 -0.5 -0.3 -0.1 0.1 0.3 0.5 z/h z/h (a) FGM II (b) FGM III Fig. 6.5 Material compositions of FGM II (a) and FGM III (b)

Table 6.2 Details of three different types of the layered FG beams

Beam type Number of layer Percentage of Al2O3 (Vc) Total thickness (h) (N) FGM I 5 layers [100-70-40-20-0] 3 mm FGM II 6 layers [100-90-70-60-40-0] 6 mm FGM III 7 layers [100-60-40-30-20-10-0] 7 mm

It is assumed that the total thickness (h) is composed of N layers of equal thickness. The volume fraction index (n) can be calculated by applying Eq. (6.1) to data from Table 6.2, for each layered beam type. The approximate volume fraction index values for FGM I, FGM II and FGM III are, n=1.3, n=0.6 and n=1.7 respectively. The predictive effectiveness of Eq. (6.1) can be seen, especially for the layered FGM composed of a higher number of layers.

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Table 6.3 The fundamental frequency (kHz) of different types of FG beams without added mass effect

L/h Material B.C. 15 20 25 30 50 C-F 1.644 0.926 0.593 0.412 0.148 FGM-I S-S 4.601 2.594 1.662 1.155 0.416 C-S 7.141 4.038 2.591 1.801 0.650 C-C 10.280 5.833 3.748 2.609 0.942 C-F 1.024 0.577 0.369 0.257 0.092 FGM-II S-S 2.863 1.615 1.035 0.719 0.259 C-S 4.437 2.512 1.612 1.122 0.405 C-C 6.375 3.624 2.331 1.623 0.587 C-F 0.630 0.355 0.227 0.158 0.057 FGM-III S-S 1.764 0.995 0.637 0.443 0.160 C-S 2.739 1.548 0.993 0.691 0.247 C-C 3.602 2.238 1.438 1.000 0.361

The fundamental frequencies (kHz) of three different types of the layered FG beams, with the different boundary conditions, ignoring the effect of any added mass, are tabulated in Table 6.3. Various layered FG beams were defined by varying the values of the thickness ratio (L/h). It can be seen that increasing the thickness ratio leads to frequency reduction, under every boundary condition. The highest frequencies are obtained from the C-C beams, followed by C-S and S-S beams respectively. The C-F beams have the lowest fundamental frequencies at every thickness ratio.

Table 6.4 Frequency results (kHz) of the layered FG beams with added mass effect

FGM-I FGM-II FGM-III B.C. L/h f1 f2 f3 f1 f2 f3 f1 f2 f3 20 0.907 5.352 15.656 0.571 3.438 9.838 0.352 2.132 6.074 C-F 25 0.583 3.479 10.176 0.366 2.227 6.357 0.226 1.378 3.923 30 0.406 2.443 7.126 0.255 1.558 4.465 0.156 0.963 2.741 20 2.405 10.201 21.635 1.552 6.382 13.698 0.960 3.938 8.485 S-S 25 1.562 6.583 14.073 1.002 4.108 8.912 0.619 2.532 5.512 30 1.096 4.590 9.925 0.700 2.863 6.334 0.432 1.763 3.841 20 3.733 12.694 25.502 2.410 7.967 15.973 1.494 4.899 8.107 C-S 25 2.429 8.222 16.607 1.559 5.148 10.459 0.965 3.181 6.492

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30 1.706 5.748 11.702 1.090 3.599 7.379 0.674 2.218 4.548 20 5.320 15.653 29.218 3.449 9.786 18.205 2.144 6.075 11.402 C-C 25 3.474 10.156 18.729 2.239 6.354 11.773 1.389 3.922 7.462 30 2.446 7.130 13.346 1.567 4.418 8.444 0.972 2.348 5.214

In Table 6.4, the first three-modes of frequencies in unit of kHz (f1,f2,f3) for the FGM I, FGM II and FGM III beams associated with several types of boundary condition are presented with different values of thickness ratio. The added mass is taken into consideration in this study. A 10 mm long mass is located at the middle of the layered FG beams. The mass magnitude per unit area is fixed as a half of the beams by using β=0.5. The resulting frequency changes due to boundary condition effects follow the same trend as the previous case, in which the highest frequencies are seen on C-C beams and the lowest frequencies on C-F beams. Once again, the frequencies decrease as the thickness ratio increases.

Table 6.5 Frequency results (kHz) of the layered FG beams with various mass ratios (L/h=20)

FGM-I FGM-II FGM-III β B.C. f1 f2 f3 f1 f2 f3 f1 f2 f3 C-F 0.915 5.500 15.753 0.573 3.492 9.849 0.353 2.160 6.081 0.3 S-S 2.476 10.234 22.108 1.577 6.386 13.829 0.974 3.939 8.553 C-S 3.847 12.781 25.817 2.449 7.990 15.978 1.515 4.942 9.920 C-C 5.509 15.434 29.642 3.516 9.793 18.384 2.180 6.076 10.721 C-F 0.908 5.352 15.654 0.571 3.439 9.833 0.352 2.132 6.075 0.5 S-S 2.405 10.178 21.846 1.552 6.381 13.672 0.961 3.939 8.435 C-S 3.733 12.693 25.268 2.409 7.968 15.911 1.494 4.927 9.835 C-C 5.321 15.653 29.180 3.450 9.802 18.366 2.144 6.073 11.391 C-F 0.897 5.158 15.513 0.568 3.366 9.812 0.350 2.092 6.062 0.8 S-S 2.311 10.156 21.102 1.518 6.376 13.388 0.942 3.936 8.323 C-S 3.580 12.569 24.967 2.354 7.933 15.494 1.464 4.914 9.782 C-C 5.070 15.540 28.602 3.356 9.774 17.938 2.093 6.068 11.223 C-F 0.891 5.046 15.541 0.566 3.320 9.824 0.349 2.067 6.064 1.0 S-S 2.253 10.126 20.739 1.496 6.373 13.295 0.931 3.937 8.270 C-S 3.485 12.463 20.938 2.319 7.912 15.579 1.445 4.905 9.761 C-C 4.921 15.465 28.306 3.297 9.204 17.540 2.061 6.064 11.071

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The impact of mass magnitude on natural frequencies is presented in Table 6.5, produced by varying the density parameter β from 0.3 to 1.0. β=1.0 means that the mass per unit area is equivalent to that of the beam. The added mass, which has the same length as the study in Table 6.4, is fixed at the middle of the beam. It can be seen that frequency decreases as the added mass increases, for every frequency mode and for every boundary condition. It can also be seen that FGM I beams gave higher frequency results than those obtained from FGM II and FGM III beams

1 (a)

0.99

0.98

f1 S-S 0.97

0.96

0.95 0 0.2 0.4 0.6 0.8 1 xm/L 2.25 (b) 2.23

2.21

2.19 f1 C-C 2.17

2.15

2.13 0 0.2 0.4 0.6 0.8 1 xm/L

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1.55 (c) 1.54 1.53 1.52 f1 1.51 C-S 1.5 1.49 1.48 0 0.2 0.4 0.6 0.8 1 xm/L 0.36 (d) 0.355

0.35

f1 0.345 C-F

0.34

0.335 0 0.2 0.4 0.6 0.8 1 xm/L

Fig. 6.6 The fundamental frequency (kHz) of FGM III beams with the effect of added mass position (L/h=20, β=0.5, d=10 mm): (a) S-S (b) C-C (c) C-S and (d) C-F

The influence of the added mass and its position on the fundamental frequencies of beam is illustrated in Fig 6.6. It is evident that the placement of added mass at the middle of the beam resulted in the lower frequencies for S-S (6.6 a) and C-C (6.6 b) beams, whereas the frequency values increased symmetrically as the mass is moved closer towards the supports. However, the C-F (6.6 d) beam has its lowest frequency when the mass is positioned close to the tip (the free condition). As expected, the C-S (6.6 c) beam has a different trend in frequency values as the mass is moved from one end to the other.

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Chapter 6

6.4 Free vibration analysis of continuous FG beams made of ceramic/polymer

This section discusses the free vibration analysis of FG beams made of general ceramic and polymer whose material properties are assumed to vary continuously throughout the beam thickness according to the power law distribution. The improved TSDT is applied to solve the vibration problem of the FG beams without including added mass, for all of the common boundary conditions. In this section, the material properties of general ceramic and polymer are assumed as follows:

3 Ceramic: Ec = 380 GPa, νc = 0.23, ρc = 3800 kg/m ,

3 Polymer: Ep = 5 GPa, νp= 0.23, ρp = 1200 kg/m .

It is assumed that the FG beams are composed of ceramic at the top surface and polymer at the bottom surface. The frequency results are presented in the form of dimensionless frequencies throughout this section, which can be expressed as the following form:

(6.7) ଶ ܫ଴ ȳൌ߱ܮ ௛ Ǥ ඩ ଶ ଶ ሺݖሻ݀ݖܧ ׬ି௛ ݄ ଶ

The material volume fraction of ceramic across the beam thickness with different values of the power law of the volume fraction index (n) is shown in Fig. 6.7. As mentioned earlier, the area under the smooth graph accounts for the percentage of ceramic in each value of the index. Thus, it is seen that increasing the value of the index leads to a reduction in the percentage of ceramic. The variations of Young’s modulus (E) and density (ρ) of ceramic/polymer FG beams with different values of n are illustrated in Fig. 6.8.

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n=0

n=0.1 Volume fraction of ceramic

n=0.5

n=1.0

n=5.0 n=10.0

z Fig. 6.7 Volume fraction of ceramic across the beam thickness with different values of the volume fraction index

176

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(a) Young’s modulus(E) Pa

(b) Density ( ρ ) Kg/m ) 3

Fig. 6.8 Material properties of FG beams across the thickness with different values of the volume fraction index; (a) Young’s modulus (Pa), (b) density (Kg/m3)

177

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Fig. 6.9 presents the dimensionless fundamental frequencies ሺȳሻ obtained from three different theories CBT, FSDT and TSDT of FG beams supported by simply supported at both ends. The FG beams are made from ceramic and polymer with n=10.0. Using the power law index n>>1.0, this refers to a high flexible FG beam because the percentage of ceramic is much less than that of polymer. For the smaller values of slenderness or thickness ratio (L/h=5-15) which can be defined as the thick beams, the TSDT shows the significant consideration to predict the frequency results in comparison with the others. Increasing the number of thickness ratio to be thin beams, the prediction between TSDT and FSDT is not too far different. However, the prediction of CBT shows the over- estimated frequency results because this theory does not take shear deformable effect into account. Therefore, it can be said that for the thick and high flexible FG beams the TSDT plays an important role in the prediction of the frequency results.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig. 6.9 Dimensionless fundamental frequencies ቆȳ ൌ ߱ ceramic/polymer FG beam with n=10.0

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The numerical results for the first three dimensionless frequencies obtained from the improved TSDT for the ceramic/polymer FG beams with different boundary conditions S- S, C-F and C-C are presented in Table 6.6. The thickness ratio (L/h=20) is used with three different values of the power law index n=0.5, 1.0 and 5.0.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of ceramic/polymer FG ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮTable 6.6 Dimensionless frequencies ቆȳ ൌ ߱ beams with L/h = 20 based on TSDT

B.C. Mode n=0.5 n=1.0 n=5.0

Ωଵ 2.656 2.413 1.724 S-S Ωଶ 10.510 9.564 6.826 Ωଷ 23.265 21.152 15.066

Ωଵ 0.948 0.861 0.615 C-F Ωଶ 5.884 5.351 3.820 Ωଷ 16.237 14.791 10.526

Ωଵ 5.961 5.425 3.883 C-C Ωଶ 16.174 14.742 10.579 Ωଷ 24.246 24.143 20.274

The dimensionless fundamental frequencies of the ceramic/polymer FG beams with n=2.0 in relation to different thickness ratios (L/h) are presented in table 6.7. Increasing the ratio leads to a rise in the number of the dimensionless fundamental frequencies for all types of beams with different support conditions.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮTable 6.7 Dimensionless fundamental frequencies ቆȳ ൌ ߱ ceramic/polymer beams with n = 2.0 based on TSDT

B.C. L/h 5 10 20 30 50 S-S 1.980 2.044 2.062 2.065 2.067 C-F 0.721 0.733 0.735 0.735 0.736 C-C 4.119 4.519 4.643 4.668 4.680 C-S 2.968 3.157 3.211 3.222 3.227 S-F 2.996 3.165 3.214 3.223 3.228 F-F 4.188 4.543 4.649 4.672 4.681

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Fig. 6.10 presents the dimensionless fundamental frequencies of ceramic/polymer FG beams with all the different end supports. The thickness ratio (L/h=15) is selected to use in this figure while the power law index (n) is varied. It is observed that the fundamental frequencies decline when the number of n increases. Moreover, the frequencies of C-F FG beams are the lowest in comparison with the frequencies from other types of end support beams for every power law index (n). It can be seen that the similarities in the frequency results are found in the cases of C-C and F-F, including C-S and S-F boundary conditions.

Τ ଶට ଶ ௛ଶ ሺ ሻ ݖ ݀ݖቇ of ܧ ଴ൗ݄ ׬ି௛Τ ଶܫ ܮFig 6.10 Dimensionless fundamental frequencies ቆȳ ൌ ߱ ceramic/polymer FG beams with various common boundary conditions

180

Chapter 7 Conclusions

In this research, the improved third order shear deformation (TSDT) is used to investigate thermal buckling and elastic vibration of functionally graded (FG) beams and plates. The Ritz method is applied to solve the theoretical formulations based on the TSDT of FG beams and plates with various boundary conditions, using the polynomial and trigonometric functions as the admissible displacements and rotations. By using the appropriate functions that must satisfy at least essential boundary conditions, this can yield accurate solutions with minimal computational effort. Available results from the open literature in some cases of thermal bucking and elastic vibration of FG beams and plates are used to validate and confirm an accuracy of the present results with excellent agreement. The new results of thermal buckling and vibration analysis of FG beams and plates are presented and discussed in detail for future investigators. To clearly understand about functionally graded materials and their vibration behaviour, a multi-step sequential infiltration technique is adopted to manufacture functionally graded specimens for analysing their microstructure and experiments are conducted to test their vibration characteristics.

According to the investigated results in this research, the concluding remarks are made as follows:

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7.1 Thermal buckling and elastic vibration of FG beams

Thermal buckling analysis of FG beams supported by different combinations of immovable boundary conditions, that are hinged-hinged (H-H), hinged-clamped (H-C) and clamped- clamped (C-C), are chosen to find out critical buckling temperature. Two types of solutions, namely temperature independent material property solution (Solution I) and temperature dependent material property (Solution II) are implemented to obtain the analytical results. To study an influence of material compositions on behaviour of FG beams and plates, three types of ceramics such as Al2O3, Si3N4, and ZrO2 which are commonly used in ceramic structural fields, are mainly selected to mix with metal of SUS304 to make FGMs. Therefore, the significant conclusions of thermal buckling analysis of such beams can be explained as follows:

¾ The Solution II using temperature dependent material properties plays an important role in the prediction of the buckling temperature of FG beams for every boundary condition, especially for thick beams which have low values of the thickness ratio (L/h≤20).

¾ There are considerable differences between buckling temperatures of FG beams obtained from the Solution I and II. However, the differences decrease gradually when the beams become thinner with the increase of the thickness ratio. It is also

found that using FG beams made of Al2O3/SUS304 and Si3N4/SUS304 produces

Solution II results with a few number of iterations, whereas, ZrO2/SUS304 beams

require many more iterations. This is because ZrO2 is quite sensitive when

temperature changes, compared to Al2O3 and Si3N4. For instance, with the increase

of temperature ΔT= 200 K, the thermal expansion coefficient ሺߙሻ of Al2O3 changes -6 -6 from 7.203x10 (1/K) to 7.454x10 (1/K), while for ZrO2 it changes from 18.591x10-6 (1/K) to 35.247x10-6 (1/K).

¾ Among three different pairs of materials used to produce FG beams, it is found that

the pair of Al2O3/SUS304 beams gives the highest buckling temperature, followed

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by the pairs of Si3N4/SUS304 and ZrO2/SUS304 beams respectively. However, a

pure ZrO2 beam has a lower buckling temperature than that of an SUS304 beam.

Therefore, it can be said that ZrO2 is one type of ceramic that could be not recommended to mix with SUS304 to produce FG beams due to its inferior material property to protect SUS304.

¾ Boundary condition is one of the most important aspects that leads to a considerable change in the buckling temperature result of the FG beam. The FG beam supported by clamped-clamped (C-C) boundary condition gives the greatest buckling temperature, followed by clamped-hinged (C-H) and hinged-hinged (H-H) boundary conditions for every thickness ratio. In addition, due to the variation of thickness ratio, the resulting buckling temperatures are reduced with the increase of the thickness ratio.

For vibration analysis of FG beams, the conclusions can be summarised as follows:

¾ To predict natural frequencies of FG beams under thermal environment, the frequency results obtained from the Solution II also show the significant differences when compared to those obtained from the Solution I. The differences between the frequency results of the Solution I and II have increased as the temperature increases.

¾ As temperature increased, it was found that the natural frequencies of FG beams decrease to the global minimum point (closing to zero) at the critical temperature or the buckling temperature. However, the trend of frequency changes is reversed when the temperature is increased beyond the critical temperature. In general, the frequency results predicted by the Solution II are lower than those of the Solution I within the range of pre-buckling region.

¾ Thick FG beams with a low value of the thickness ratio (L/h) have higher natural frequencies than those of thinner beams for every boundary condition. Obtaining the critical temperature using vibration analysis, can be achieved by varying the

183

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values of the thickness ratio and the material volume fraction index (n) as well. The given temperature can account for the critical temperature if it causes the minimum value of the fundamental frequency of the FG beam.

¾ In order to understand more behaviour of FG beams subjected to dynamic load and to verify theoretical formulations, FG beam specimens are fabricated using a multi- step sequential infiltration technique for vibration testing in this research. The technique used to produce alumina-epoxy FG beam samples shows its effectiveness to fabricate good quality samples with a few porosities. In theoretical analysis, the material compositions of FG beams have often been assumed to vary smoothly and continuously across the graded direction. However, the various manufacturing techniques including the one presented in this research which are used for fabricating FG beams cannot easily produce them with properties varying smoothly according to a continuous mathematical function. Rather, existing techniques produce layered FG beams. Therefore, to obtain effective material properties of the layered FG beams, the power law distribution is modified for calculating the approximate power law index. It is revealed that the approximated material composition profiles match well with the real profiles of the layered FG beams.

¾ For vibration testing of the layered FG beams, the experimental results are compared to the theoretical results with reasonable agreement. The discrepancies between theoretical and experimentally observed frequencies are around 10% for every boundary condition. Imperfect epoxy infiltration leading to porosities, during the FGM manufacturing process, and error due to approximation of the material composition profile are sufficient explanation for these discrepancies.

¾ To consider added mass effect on frequencies of FG beams, it is found that increased added mass leads to reduction in frequencies for every boundary condition. Additionally, the effect of the position of the added mass has a significant impact on frequencies as well. The placement of added mass at the middle of the beam results in the lower frequencies for simply supported-simply

184

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supported (S-S) and clamped-clamped (C-C) beams, whereas the frequency values increase symmetrically as the mass is moved closer towards the end supports. However, for the case of clamped-free (C-F) beam, its lowest frequency is found when the mass is placed at the tip (the free condition). For another beam which is the clamped-simply supported (C-S) beam, the trend of frequency changes due to the mass being moved from one end to the other is different compared to other beams.

7.2 Thermal buckling and elastic vibration of FG plates Based on the investigated results, the conclusions of thermal buckling analysis of FG plates are presented as follows:

¾ By using the improved TSDT for predicting thermal buckling results of FG plates, the theory reveals its significance when it is used to deal with thick FG plates (b/h≤20). This is due to the improved terms of the theory.

¾ Owing to the existence of bending-coupling and warping-extensional coupling effects of FG plates, the buckling temperature of the immovable simply supported FG plate is different from that of the movable one. However, due to the vanishing of the coupling effects for isotropic plate, the buckling temperature of the plate supported by movable or immovable simply supported presents the same result.

¾ Similarly with the investigation on FG beams, the Solution II plays an important role in the prediction of buckling temperatures of simply supported and clamped FG plates. The differences between the results of the Solution I and II are greater when the FG plates become thicker. Generally, the clamped FG plate gives a much higher buckling temperature result than that obtained from the simply supported FG plate.

¾ Increasing the value of the volume fraction index (n) leads to a reduction in the

buckling temperature of FG plates made of Al2O3/SUS304 and Si3N4/SUS304, 185

Chapter 7

whereas, the buckling temperature of the ZrO2/SUS304 plate increases as the increment of value n. It is also observed that the buckling temperature of the

SUS304 plate is higher than that of the ZrO2 plate. Thus ZrO2 is the ceramic that could not be recommended to mix with SUS304 to produce an FG plate.

¾ Based on the numerical results, this reveals that the highest buckling temperature is

obtained from the Al2O3/SUS304 plate followed with Si3N4/SUS304 and

ZrO2/SUS304 plates respectively for every value of plate aspect ratio and volume fraction index as well as boundary condition.

In terms of thermo-elastic free and forced vibration analysis of FG plates, the concluding remarks are made as:

¾ Natural frequencies of FG plates made of any pair of materials are reduced by increasing the value of the volume fraction index (n).

¾ By increasing temperature, the frequency changes of FG plates follow the same trend of the changes in FG beam analysis in which the frequency decreases as temperature increases. The global minimum frequency is detected at the critical temperature. The increasing trend of frequency changes is found in the post- buckling region.

¾ The natural frequencies of FG plates under thermal environment which are detected in the case of a clamped boundary condition are much higher than those obtained from the case of simply supported.

¾ By considering forced vibration response of FG plates, deflections due to uniformly distributed dynamic loads have changed slightly at the zone far from the resonance

point (ݎఠ ൌͳǤͲ) and they approach infinity when the frequency of the external excitation corresponds to the natural frequency of the system. The dramatic increase of deflections is found around the zone of resonance. Because of the bending

stiffness of the ZrO2/SUS304 plate being weaker than that of the Al2O3/SUS304 and

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Chapter 7

Si3N4/SUS304 plates, therefore, the deflection of the ZrO2/SUS304 plate is too high compared to other plates.

¾ Additionally, increasing the values of the volume fraction index and temperature lead to higher deflection of the plate. As expected, the deflection of the clamped FG plate is lower than that of the simply supported plate.

7.3 Further work

For further work on this area of research, there are several subjects that can be extended for investigation in the future development.

¾ The main emphasis of this research was on the linear analysis of FG beams and plates using the improved TSDT. It is believed that the presented formulations in this research are useful and sufficient for designing the beam and plate structures made from functionally graded materials, and they can be considered as the useful background for the future development of FG beam and plate analysis. In case of FG structures subjected to the severe static and dynamic loadings, the structures usually exhibit large deformations. Thus the main focus to deal with the problems is to further study the nonlinear behaviour of the FG structures. The displacement field of the improved TSDT can be used with the von Karman type nonlinear strain-displacement relationships in order to construct the nonlinear governing equations.

¾ The mathematical formulations based on the improved TSDT can be implemented in a finite element framework for developing FGM shape functions and elements. As a result, the complex structures made from FGMs can be solved easily with this theory.

¾ FG beam specimens were successfully manufactured using the multi-step sequential infiltration technique. It is expected that the ideas and principles of the effective technique presented in this research can be used and applied to produce a 187

Chapter 7

big panel of FG plate specimen, for mechanical experiments such as bending, buckling and vibration tests. However, to produce the bigger specimens, the bigger size of laboratory equipment such as compressing foam spacer, vacuum chamber, furnace etc are required. In the future, the infiltration technique will be modified and then conducted to manufacture FG specimens again, using metal foam instead of polyurethane foam in order to produce metal-ceramic specimens. By using the idea, it is important to select ceramics that can be solidified by using not too high temperature. Because if using too high temperature for the sintering process, this leads to the shrinkage between adjacent compositions of metal and ceramic phases, or another problem is that the metal foam might be burnt out. It would be aware that the metal phase is immigrated obviously during sintering due to very high temperature whereas it is quite low for the ceramic phase.

¾ In this current research, the FG beam specimens were tested for investigating their vibration behaviour under ambient temperature. Hence, it is interesting to extend the vibration test of the specimens under thermal environment in the future investigation. The vibration experimental set up needs more equipment such as heater, temperature controller and accelerometer that can measure the signal at thermal environment. Therefore, the experimental results of thermo-elastic vibration can be compared and validated with the theoretical results that can be computed using the formulations provided in this research.

188

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