Thermal buckling and elastic vibration analysis of functionally graded beams and plates using improved third-order shear deformation theory A thesis By 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. Signed ……………………………………………………………… Date …16/07/2012…………………………………………………… ii Copyright Statement I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or hereafter known, subject to the provisions of the Copyright Act 1968. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the abstract of my thesis in Dissertations Abstract International (this is applicable to doctoral theses only). I have either used no substantial portions of copyright material in my thesis or I have obtained permission to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation. Signed ……………………………………………………………… Date ……16/07/2012……………………………………………… iii Authenticity Statement 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. Signed ……………………………………………………………… Date …16/07/2012………………………………………………… iv 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. vi 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. vii 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 viii 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 ix 4.2.2 Thermo-elastic