Carbon Nanotubes and Nanosensors Vibration, Buckling and Balistic Impact

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.

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

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