Carbon Nanotubes and Nanosensors: Vibration, Buckling and Balistic Impact

The main properties that make carbon nanotubes (CNTs) a promising technology for many future applications are: extremely high strength, low mass density, linear elastic behavior, almost perfect geometrical structure, and nanometer scale structure. Also, CNTs can conduct electricity better than copper and transmit heat better than diamonds. Therefore, they are bound to find a wide, and possibly revolutionary use in all fields of engineering. The interest in CNTs and their potential use in a wide range of commercial applications; such as nanoelectronics, quantum wire interconnects, field emission devices, composites, chemical sensors, biosensors, detectors, etc.; have rapidly increased in the last two decades. However, the performance of any CNT-based nanostructure is dependent on the mechanical properties of constituent CNTs. Therefore, it is crucial to know the mechanical behavior of individual CNTs such as their vibration frequencies, buckling loads, and deformations under different loadings. This title is dedicated to the vibration, buckling and impact behavior of CNTs, along with theory for carbon nanosensors, like the Bubnov-Galerkin and the Petrov-Galerkin methods, the Bresse-Timoshenko and the Donnell shell theory.

Table of Contents:
Preface xi Chapter 1. Introduction 1 1.1. The need of determining the natural frequencies and buckling loads of CNTs 8 1.2. Determination of natural frequencies of SWCNT as a uniform beam model and MWCNT during coaxial deflection 8 1.3. Beam model of MWCNT 9 1.4. CNTs embedded in an elastic medium 10 Chapter 2. Fundamental Natural Frequencies of Double-Walled Carbon Nanotubes 13 2.1. Background 13 2.2. Analysis 15 2.3. Simply supported DWCNT: exact solution 15 2.4. Simply supported DWCNT: Bubnov–Galerkin method 18 2.5. Simply supported DWCNT: Petrov–Galerkin method 20 2.6. Clamped-clamped DWCNT: Bubnov–Galerkin method 23 2.7. Clamped-clamped DWCNT: Petrov–Galerkin method 25 2.8. Simply supported-clamped DWCNT 27 2.9. Clamped-free DWCNT 30 2.10. Comparison with results of Natsuki et al. [NAT 08a] 33 2.11. On closing the gap on carbon nanotubes 34 2.12. Discussion 45 Chapter 3. Free Vibrations of the Triple-Walled Carbon Nanotubes 47 3.1. Background 47 3.2. Analysis 48 3.3. Simply supported TWCNT: exact solution 49 3.4. Simply supported TWCNT: approximate solutions 51 3.5. Clamped-clamped TWCNT: approximate solutions 54 3.6. Simply supported-clamped TWCNT: approximate solutions 57 3.7. Clamped-free TWCNT: approximate solutions 60 3.8. Summary 63 Chapter 4. Exact Solution for Natural Frequencies of Clamped-Clamped Double-Walled Carbon Nanotubes 65 4.1. Background 65 4.2. Analysis 67 4.3. Analytical exact solution 72 4.4. Numerical results and discussion 77 4.5. Discussion 82 4.6. Summary 83 Chapter 5. Natural Frequencies of Carbon Nanotubes Based on a Consistent Version of Bresse–Timoshenko Theory 85 5.1. Background 85 5.2. Bresse–Timoshenko equations for homogeneous beams 86 5.3. DWCNT modeled on the basis of consistent Bresse–Timoshenko equations 88 5.4. Numerical results and discussion 91 Chapter 6. Natural Frequencies of Double-Walled Carbon Nanotubes Based on Donnell Shell Theory 97 6.1. Background 97 6.2. Donnell shell theory for the vibration of MWCNTs 99 6.3. Donnell shell theory for the vibration of a simply supported DWCNT 100 6.4. DWCNT modeled on the basis of simplified Donnell shell theory 103 6.5. Further simplifications of the Donnell shell theory 105 6.6. Summary 107 Chapter 7. Buckling of a Double-Walled Carbon Nanotube 109 7.1. Background 109 7.2. Analysis 111 7.3. Simply supported DWCNT: exact solution 112 7.4. Simply supported DWCNT: Bubnov–Galerkin method 114 7.5. Simply supported DWCNTs: Petrov–Galerkin method 116 7.6. Clamped-clamped DWCNT 117 7.7. Simply supported-clamped DWCNT 119 7.8. Buckling of a clamped-free DWCNT by finite difference method 121 7.9. Buckling of a clamped-free DWCNT by Bubnov–Galerkin method 131 7.10. Summary 137 Chapter 8. Ballistic Impact on a Single-Walled Carbon Nanotube 139 8.1. Background 139 8.2. Analysis 140 8.3. Numerical results and discussion 144 Chapter 9. Clamped-Free Double-Walled Carbon Nanotube-Based Mass Sensor 149 9.1. Introduction 149 9.2. Basic equations 150 9.3. Vibration frequencies of DWCNT with light bacterium at the end of outer nanotube 152 9.4. Vibration frequencies of DWCNT with heavy bacterium at the end of outer nanotube 159 9.5. Vibration frequencies of DWCNT with light bacterium at the end of inner nanotube 165 9.6. Vibration frequencies of DWCNT with heavy bacterium at the end of inner nanotube 170 9.7. Numerical results 176 9.8. Effective stiffness and effective mass of the double-walled carbon nanotube sensor 178 9.9. Virus sensor based on single-walled carbon nanotube treated as Bresse–Timoshenko beam 190 9.10. Conclusion 201 Chapter 10. Some Fundamental Aspects of Non-local Beam Mechanics for Nanostructures Applications 203 10.1. Background on the need of non-locality 204 10.2. Non-local beam models 209 10.3. The cantilever case: a structural paradigm 218 10.4. Euler–Bernoulli beam: Eringen’s based model 231 10.5. Euler–Bernoulli beam: gradient elasticity model 234 10.6. Euler–Bernoulli beam: hybrid non-local elasticity model 236 10.7. Timoshenko beam: Eringen’s based model 238 10.8. Timoshenko beam: gradient elasticity model 244 10.9. Timoshenko beam, hybrid non-local elasticity model 251 10.10. Higher order shear beam: Eringen’s based model 254 10.11. Higher order shear beam, gradient elasticity model 259 10.12. Validity of the results for double-nanobeam systems 262 Chapter 11. Surface Effects on the Natural Frequencies of Double-Walled Carbon Nanotubes 269 11.1. Background 269 11.2. Analysis 271 11.3. Results and discussion 279 11.4. Surface effects on buckling of nanotubes 286 11.5. Summary 289 Chapter 12. Summary and Directions for Future Research 291 Appendix A. Elements of the Matrix A 297 Appendix B. Elements of the Matrix B 299 Appendix C. Coefficients of the Polynomial Equation [7.116] 301 Appendix D. Coefficients of the Polynomial Equation [9.25] 303 Appendix E. Coefficients of the Polynomial Equation [9.35] 305 Appendix F. Coefficients of the Polynomial Equation [9.40] 307 Appendix G. Coefficients of the Polynomial Equation [9.54] 311 Appendix H. Coefficients of the Polynomial Equation [9.63] 313 Appendix I. Coefficients of the Polynomial Equation [9.67] 315 Appendix J. An Equation Both More Consistent and Simpler than the Bresse–Timoshenko Equation 319 Bibliography 325 Author Index 399 Subject Index 415

About the Author :
Prof. Isaac Elishakoff, Florida Atlantic University, USA. Dr. Demetris Pentaras, Cyprus University of Technology, Cyprus. Ing. Kevin Dujat and Ing. Simon Bucas, IFMA – French Institute for Advanced Mechanics, France. Dr. Claudia Versaci and Prof. Giuseppe Muscolino, University of Messina, Italy. Dr. Joel Storch, Touro College, USA. Prof. Noël Challamel, University of South Brittany, France. Prof. Toshiaki Natsuki, Shinsu University, Japan. Dr. Yingyan Zhang, University of Western Sydney, Australia. Prof. Chien Ming Wang, National University of Singapore, Singapore. Ing. Guillaume Ghyselinck, Ecole des Mines d’Alès, France.