Carbon Nanotubes and Nanosensors

Carbon Nanotubes and Nanosensors

Vibration, Buckling and Ballistic Impact

Isaac Elishakoff, Demetris Pentaras, Kevin Dujat Claudia Versaci, Giuseppe Muscolino, Joel Storch Simon Bucas, Noël Challamel, Toshiaki Natsuki Yingyan Zhang, Chien Ming Wang, Guillaume Ghyselinck First published 2012 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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The rights of Isaac Elishakoff, Demetris Pentaras, Kevin Dujat, Simon Bucas, Claudia Versaci, Giuseppe Muscolino, Joel Storch, Noël Challamel, Toshiaki Natsuki, Yingyan Zhang, Chien Ming Wang, Guillaume Ghyselinck to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. ______Library of Congress Cataloging-in-Publication Data

Carbon nanotubes and nanosensors : vibration, buckling, and ballistic impact / Isaac Elishakoff ... [et al.]. p. cm. Includes bibliographical references and index. ISBN 978-1-84821-345-6 1. Nanotubes--Impact testing. 2. Nanotubes--Elastic properties. 3. Detectors--Testing. I. Elishakoff, Isaac. TA418.9.N35C357 2011 620.1'15--dc23 2011043131 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-345-6

Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CR0 4YY 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.BeammodelofMWCNT...... 9 1.4.CNTsembeddedinanelasticmedium...... 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-freeDWCNT...... 30 2.10. Comparison with results of Natsuki et al. [NAT 08a]...... 33 2.11. On closing the gap on carbon nanotubes ...... 34 2.11.1.Linearanalysis...... 34 2.11.2.Nonlinearanalysis...... 40 2.12.Discussion...... 45 vi Carbon Nanotubes and Nanosensors

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-clampedTWCNT:approximatesolutions...... 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.Analyticalexactsolution...... 72 4.4.Numericalresultsanddiscussion...... 77 4.4.1.Bubnov–Galerkinmethod...... 81 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.Numericalresultsanddiscussion...... 91

Chapter 6. Natural Frequencies of Double-Walled Carbon Nanotubes Based on Donnell Shell Theory...... 97 6.1.Background...... 97 6.2.DonnellshelltheoryforthevibrationofMWCNTs...... 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 Table of Contents vii

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-clampedDWCNT...... 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 7.9.1.Analysis...... 131 7.9.2.Results...... 135 7.9.3.Conclusion...... 137 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.Numericalresultsanddiscussion...... 144

Chapter 9. Clamped-Free Double-Walled Carbon Nanotube-Based Mass Sensor ...... 149 9.1. Introduction ...... 149 9.2.Basicequations...... 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.Numericalresults...... 176 9.8. Effective stiffness and effective mass of the double-walled carbon nanotube sensor ...... 178 9.8.1.Introduction...... 178 9.8.2.Bubnov–Galerkinmethod...... 179 9.8.3.Finite-differencemethod...... 182 9.8.4.EffectivemassofDWCNTwithbacteriumattheend.... 186 viii Carbon Nanotubes and Nanosensors

9.8.5. Conclusions ...... 190 9.9. Virus sensor based on single-walled carbon nanotube treated as Bresse–Timoshenko beam...... 190 9.9.1.Introduction...... 190 9.9.2.Analysis...... 191 9.9.3.Results...... 197 9.9.4.Mimivirus...... 200 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-localbeammodels...... 209 10.2.1.BeammechanicsandEringen’snon-localmodel...... 209 10.2.2.Beammechanicsandgradientelasticitymodel...... 212 10.2.3. How to connect gradient elasticity with non-local integral models?...... 216 10.3. The cantilever case: a structural paradigm ...... 218 10.3.1.Introduction...... 218 10.3.2.Eringen’sintegralmodel...... 219 10.3.3.Gradientelasticbeam...... 221 10.3.4. Non-local elastic beam based on strain energy functional with squared non-local curvature ...... 224 10.3.5. New non-local elastic beam model based on strain energy functional with mixed local and non-local curvatures ... 225 10.4. Euler–Bernoulli beam: Eringen’s based model ...... 231 10.4.1. Buckling of non-local Euler–Bernoulli beams ...... 231 10.4.2. Vibrations of non-local Euler–Bernoulli beams ...... 233 10.5. Euler–Bernoulli beam: gradient elasticity model...... 234 10.5.1. Buckling of gradient elasticity Euler–Bernoulli beams . . 234 10.5.2. Vibrations of gradient elasticity Euler–Bernoulli beams . 235 10.6. Euler–Bernoulli beam: hybrid non-local elasticity model ... 236 10.6.1. Buckling of hybrid non-local Euler–Bernoulli beams ... 236 10.6.2. Vibrations of hybrid non-local Euler–Bernoulli beams . . 237 10.7. Timoshenko beam: Eringen’s based model ...... 238 10.7.1. Buckling of non-local Engesser Timoshenko beams .... 238 10.7.2. Buckling of non-local Haringx Timoshenko beams .... 240 10.7.3. Vibrations of non-local Timoshenko beams ...... 242 10.8. Timoshenko beam: gradient elasticity model ...... 244 10.8.1. Buckling of gradient Timoshenko beam with Engesser’s theory ...... 244 Table of Contents ix

10.8.2. Buckling of gradient Timoshenko beam with Haringx’s theory ...... 246 10.8.3. Vibrations of gradient Timoshenko beam ...... 247 10.8.4. Some other Timoshenko gradient elasticity beam models. 249 10.9. Timoshenko beam, hybrid non-local elasticity model...... 251 10.9.1. Buckling of the hybrid Engesser’s non-local Timoshenko beam ...... 251 10.9.2. Vibrations of the hybrid non-local Timoshenko beam ... 252 10.10. Higher order shear beam: Eringen’s based model ...... 254 10.10.1.Bucklingofnon-localhigherordershearbeam...... 254 10.10.2.Vibrationsofnon-localhigherordershearbeam..... 257 10.11.Higherordershearbeam,gradientelasticitymodel...... 259 10.11.1. Buckling of gradient Engesser higher order shear beam . 259 10.11.2. Vibrations of gradient higher order shear beam ...... 260 10.12.Validityoftheresultsfordouble-nanobeamsystems..... 262 10.12.1. Buckling of non-local double-nanobeam systems ..... 262 10.12.2. Vibrations of non-local double-nanobeam systems .... 265 10.12.3.Bucklingofgradientdouble-nanobeamsystems...... 266 10.12.4. Vibrations of gradient double-nanobeam systems ..... 267

Chapter 11. Surface Effects on the Natural Frequencies of Double-Walled Carbon Nanotubes ...... 269 11.1. Background ...... 269 11.2.Analysis...... 271 11.2.1. Non-local Bresse–Timoshenko beam theory ...... 272 11.2.2.VanderWaalsinteractionforces...... 274 11.2.3.NaturalvibrationofDWCNTs...... 275 11.2.4.FreevibrationofembeddedDWCNTs...... 278 11.3.Resultsanddiscussion...... 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 x Carbon Nanotubes and Nanosensors

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 Preface

I would like to describe a field, in which little has been done, but in which an enormous amount can be done in principle. This field is not quite the same as the others … Furthermore, a point that is most important is that it would have an enormous number of technical applications. What I want to talk about is the problem of manipulating and controlling things on a small scale.

Richard Feynman

Carbon nanotubes (CNTs) were discovered by Sumio Iijima of the NEC Corporation in the early 1990s. Since then, extensive research activities on CNTs have been initiated around the world. This interest is attributed to the extraordinary mechanical properties and unique electrical properties of CNTs and their potential applications. Meyyappan [MEY 05] remarked that “the breadth of applications for carbon nanotubes is indeed wide ranging: nanoelectronics, quantum wire interconnects, field emission devices, composites, chemical sensors, biosensors, detectors, etc … The community is beginning to move beyond the wonderful properties that interested them in CNTs and are beginning to tackle real issues associated with converting a material into a device, a device into a system, and so on”. In a broader sense, Liu et al. [LIU 06] stressed that “nanotechnology is making, and will continue to make, an impact in key areas for societal improvement”.

For the reader who is new to the world of nanotechnology, we quote from the book by Rogers et al. [ROG 08]: “The ‘nano’, from which this relatively new field derives its name, is a prefix denoting 10–9. ‘Nano’ comes from xii Carbon Nanotubes and Nanosensors nanos, a Greek word meaning dwarf. In the case of nanotechnology, it refers to things in the ballpark that are one-billionth of meter in size. While in graduate school in 1905, Albert Einstein took experimental data on the diffusion of sugar in water and showed that a single sugar molecule is about one nanometer in diameter … Nobel laureates, novelists, and news anchors alike tell us on a daily basis that nanotechnology will completely change the way we live. They have promised us microscopic, cancer-eating robots swimming through our veins! Self-cleaning glass! Digital threads! Electronic paper! Palm-sized satellites! The cure for deafness! Molecular electronics: Smart dust!”.

This book deals with specific aspects of CNTs only, namely their vibrations, buckling, impact buckling, and nanosensors. For vibration and buckling analyses, we use the classical Bernoulli–Euler theory of beams. However, as it turned out recently, long CNTs may pose health risks that are similar to those found in asbestos, with possible diseases such as mesothelioma or cancer of the lining of the lungs as well as adverse effects on the male reproductive system. Since long CNTs are harmful whereas short CNTs are not, we must use the short CNTs along with the theory that is appropriate for the short CNTs. Specifically, we use Bresse–Timoshenko theory for short CNTs because when the length to diameter ratio is relatively small, transverse shear deformation and rotary inertia must be accounted for. We use a consistent and simple version of Bresse–Timoshenko theory that has been recently developed by the first author. This analysis leads to simple expressions for natural frequencies. A theory of nanosensors is presented to identify the possibility of attached virus or bacterium. Both long and short CNTs may be regarded as nanosensors.

This promise of “the next big idea of nanotechnology” virtually forces us to contribute, at least in some small manner, to the noble goals above. This book deals with CNTs. We owe our gratitude to many scientists around the world. It is our pleasure to record appreciation to several individuals with whom we discussed our findings (as listed in alphabetical order): Professor Sondipon Adhikari, University of Swansea, United Kingdom; Professor Romesh Batra and Dr. S.S. Gupta of the Virginia Polytechnic Institute and State University, USA; Professor Qing Chen of Peking University, People’s Republic of China; Professor Jean-Michel Claverie of Institut de Microbiologie de la Méditerannée, France; Professor Moshe Eisenberger of the Technion-Israel Institute of Technology, Israel; Dr. Rivka Gilat, University of Ariel, Israel; Professor Lin Guo and Professor L.D. Li of the Preface xiii

Beijing University of Aeronautics and Astronautics, People’s Republic of China; Professor George Kardomateas, Georgia Institute of Technology, USA; Professor Fred van Keulen and Professor Gary Steele of the Delft University of Technology, The Netherlands; Professor Michael Link of the Gesamthochschule Wuppertal, Germany; Professor Nicola Pugno of the Politecnico di Torino, Italy; Professor Gabor Stepan, Professor Tibor Tarnai, and Professor Lajos Pomazi of the Budapest University of Technology and Economics, Hungary; Professor X. Frank Xu of Stevens Institute of Technology, USA; last but not least our thanks go to Professor Gopal Gaonkar, Professor Theodora Leventouri, and Professor Hassan Mahfuz of the Florida Atlantic University, USA.

Naturally, none of above researchers bears any responsibility for the contents of this book. We are extremely indebted to Mr. Clément Soret of Institut Français de Mécanique Avancée for his painstaking work of introducing numerous corrections to the text that were detected by the authors, and especially by Joel Storch. We are also grateful to Mr. Yohann Miglis of the Florida Atlantic University for kindly preparing the author and subject indexes.

We will be most grateful to the readers if they will be so kind as to communicate to us their constructive comments on both the content of this multi-continental effort and on possible extensions and cooperations.

Isaac Elishakoff Boca Raton, USA Demetris Pentaras Limassol, Cyprus Kevin Dujat Aubière, France Claudia Versaci Messina, Italy Giuseppe Muscolino Messina, Italy Joel Storch Los Angles, USA Simon Bucas Aubière, France Noël Challamel Rennes, France Toshiaki Natsuki Shinsu, Japan Chien Ming Wang Singapore Yingyan Zhang Penrith, Australia Guillaume Ghyselinck Alès, France January 2012

Chapter 1

Introduction

Steam power enabled much that happened in the 19th Century. Electricity enabled much that happened in the 20th. Nanotechnology will enable much that happens in this century … There has been a lot of hype about nanotechnology.

James Marting, 2007

Nanotechnology is one of the most promising and exciting fields of sciences today.

Congressional Record, Vol. 149, p. 1.22, Nov 20, 2003

Many scientific publications attribute the discovery of carbon nanotubes (CNTs) in 1991 to Sumio Iijima. It must be emphasized that idea itself already existed in 1960, when the Nobel-prize physicist Richard Feynman predicted the possibility of innovations that will allow for “manipulating and controlling things on small-scale”. The main properties that make CNTs a promising technology for many future applications are: extremely high strength, low mass density, linear elastic behavior up to strains of 12%, almost perfect geometrical structure, and nanometer scale structure.

As Barrera and Shofner [BAR 05] mention “with a tensile modulus of the order of 1 terapascal (TPa = 1012 Pa) and tensile strength 100 times greater than steel at one sixth of weight”, CNTs are “ideal reinforcements” for 2 Carbon Nanotubes and Nanosensors various structures. 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.

As Qian et al. [QIA 02a] mentions, “The discovery of multi-walled carbon nanotubes (MWCNTs) in 1991 has stimulated ever-broader research activities in science and engineering devoted entirely to carbon nanostructures and their applications. This is due in large part to the combination of their expected structural perfection, small size, low density, high stiffness, high strength (the tensile strength of the outermost shell of MWCNT is approximately 100 times greater that that of aluminum), and excellent electronic properties. As a result, carbon nanotubes (CNTs) may find use in a wide range of application in material reinforcement, field emission panel display, chemical sensing, drug delivery, and nanoelectronics”.

Two types of CNTs exist: the single-walled carbon nanotube (SWCNT), which is a hollow cylinder of a graphite sheet, and the multiwalled carbon nanotube (MWCNT), which consists of a number of coaxial SWCNTs. Generally, “most researchers have adopted the equilibrium interlayer spacing between adjacent nanotubes (about 0.34 nm) as the representative thickness of SWCNTs combined with a Young’s modulus of about 1 TPa” [RU 04].

The academic and industrial interest in CNTs and their potential use in a wide range of commercial applications have rapidly increased in the past 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 may lead to tailoring CNTs for specific functional and operational nanostructures. According to Ru [RU 04], “most potential applications of CNTs are heavily based on a thorough understanding of their mechanical behavior. For example, experiments and molecular dynamics simulations showed that electronic properties of CNTs can be changed by deformations up to several orders of magnitude. This can explain why the study of mechanical behavior of CNTs has been one topic of major concern”.

Besides experiments and molecular dynamics (MD), continuum mechanics has been used for understanding the mechanical behavior of CNTs. Continuum mechanics is a branch of mechanics that deals with the Introduction 3 mechanical behavior of material modeled as a continuum. We may wonder if continuum mechanics is at all applicable to CNTs. According to Yakobson and Smalley [YAK 97], the behavior of the hollow tiny tubes is more complex than rods or columns, but still predictable with continuum-elastic methods. In 1996, Yakobson et al. [YAK 96] showed, using MD simulations, that all the buckling patterns displayed by the MD simulations could be predicted by the continuum shell model. Subsequently, continuum models have been used to study the mechanical behavior of CNTs by a number of researchers.

It can be seen from a search of the literature that most experiments and MD simulations are focused on the investigation of the mechanical properties of SWCNTs (e.g. [TRE 96, PON 99, WAN 01, AGR 06], and [BAT 08]), and a limited number of works have been carried out on the vibration, buckling, and impact of CNTs, especially for MWCNTs (e.g. [ZHA 08e]) because of their relatively complex structure and large atomic size. He et al. [HE 04] in their study on buckling analysis of MWCNTs mentioned that in their MD simulation of buckling behavior, “the calculation for a SWCNT (10, 10) CNT with 2000 atoms would have required 36 h of computational time on the SGI Origin 2000 system. The computational time increases as the number of atoms increases, and thus the computational time explodes exponentially for MWCNTs”. The authors also quote a study [LIE 04] of a 4-walled CNT under tension with length to diameter ratio of 9.1 containing 15,097 atoms. The calculation for the elasto-plastic deformation up to failure took up over two months on the SGI Origin 3000 system.

A similar remark on MD simulation of MWCNTs was made by Wang and Varadan [WAN 6b]: “MD stimulation is an effective research technique which simulates accurately the physical properties of structures at the atomic-scale level. However, the computational problem here is that the time steps involved in MD simulations are limited by the vibration modes of the atoms of the order of femto-seconds (10−15). Therefore, even after a million time steps, we can reach only the nanosecond range, during which period most of the mechanical, physical or magnetic events would not even have started”.

In addition to the fact that currently MD is time consuming and not yet able to simulate large atomic scale, it is also “computationally expensive for 4 Carbon Nanotubes and Nanosensors large-sized atomic systems” as stated by Yoon et al. [YOO 02, YOO 03], Xia et al. [XIA 04], Ru [RU 04], Zhang et al. [ZHA 05a], Wang and Varadan [WAN 06b], Natsuki et al. [NAT 08a], and Xu et al. [XU 08b], to mention just a few.

Also, Dr. Kardomateas in a private communication (January 29, 2009) to one of us, wrote that “molecular dynamics studies versus beam models have not been done yet [extensively] to the best of my knowledge. It is a good research topic, one can even write a research proposal on this. Regarding experiments, I know some Japanese scientists were proposing to do experiments (I reviewed their proposal)” but he is not aware of any published results.

It must be stressed that the number of molecular mechanics-based studies is on a continuous and steep increase. A recent paper by Gupta et al. [GUP 09b] is tellingly entitled “Breakdown of Structural Models for Vibrations of Single-Wall ZigZag Carbon Nanotubes”. They conclude that “a continuum structure cannot represent all modes of vibration of a zigzag SWCNT”. They also stress:

Suffice it to say that almost all works on SWCNTs have either assumed a value for the wall thickness, or have derived it by equating an experimental observation to that deduced from a structural mechanics model.

Also,

… not all deformations of SWCNT can be studied from those of its ECS (Equivalent Continuum Structure).

They could be explained by the fact that

… with an increase in the circumferential wave number, the frequencies of the inextensional modes of vibration of the shell and the cylinder monotically increase, but those of the corresponding SWCNT saturate to a steady value.

The main conclusion of the definitive study by Gupta et al. [GUP 09b] is that their Introduction 5

… study provides an example of class of deformations in which the responses of zigzag tubes differ from those of armchair and chiral SWCNTs, and also points to the need of providing limits of applicability of a structural model of a SWCNT.

These intriguing findings need analysis and interpretation. Naturally, natural frequencies (pun intended!) are needed to compute responses of the system to specific excitations. Therefore, if the excitation is such that the contribution of saturating frequencies (accurately evaluated by MD simulations but not by its equivalent continuum structure) is important, we must conclude that the equivalent continuum structural analysis fails or breaks down. However, if the excitation is such that the excited frequencies (i.e. frequencies that contribute to the response that is evaluated by normal mode analysis) are outside the region of saturation, the continuum model can safely be used, despite the fact that saturating frequencies are not accurately predicted by the continuum modeling. In these circumstances, it appears preferable that the notion of “breakdown of structural models for vibrations of single-wall zigzag carbon nanotubes” be substituted by a more cautions statement on “possible breakdown of structural models for vibrations of single-walled zigzag carbon nanotubes for some type of excitations”. Naturally, more research is needed (funding agencies: where are you?) in order to delimit the legitimate regions where the continuum model can be used and where it fails; indeed, according to Albert Einstein, “the theory must be as simple as possible but not simpler”. The definite study by Gupta et al. [GUP 09b] may be interpreted as the continuum model sometimes being an oversimplification. In other cases, however, it appears that the continuum model is adequate.

Several works are reported in the literature on the mechanical behavior of MWCNTs by using continuum models that might be applicable for the analysis of MWCNTs as they are successfully used for the analysis of SWCNTs (e.g. [RU 04] and section 1.2 of this book). Besides the fact that continuum models offer a cost-effective alternative method to analyze MWCNTs, they also help researchers on this topic to guide and understand experiments. These works in the literature are briefly described and cited in the following chapters of this book. 6 Carbon Nanotubes and Nanosensors

SWCNTs compared to MWCNTs have higher specific stiffness and strength, but MWCNTs have higher resistance to bending and buckling than SWCNTs. Although there was some work conducted on vibration and buckling of MWCNTs, no author paid attention on increased modal density of eigenvalues and its crucial influence on buckling and vibrations of them. Modal density is the number of eigenvalues per unit of frequency (for vibrations) or load (for buckling). Indeed, the double-walled carbon nanotube (DWCNT) has two series of infinite frequencies, nearly doubling the modal density at high frequency range in comparison to its SWCNT counterpart. Likewise, an N-walled CNT has N series of natural frequencies, with attendant nearly N-tuple increase of the modal density.

Ru [RU 04] stresses that “MWCNTs are different from traditional elastic beams due to their hollow multilayer structure and the associated interlayer van der Waals forces”. The exact solutions lead to the need of solving transcendental equations. It appears that they can be usefully supplemented by simple solutions.

This book is dedicated to the vibration, buckling, and impact behavior of CNTs, along with the theory for carbon nanosensors. Whereas our main focus is on obtaining explicit formulas of natural frequencies and buckling loads for the MWCNTs at different boundary conditions, we also focus on derivation of simple expressions for the dynamic deflections of SWCNTs under impact loading (Chapter 8).

The significant features of research reported in this book are the derivations of approximate solutions by using Bubnov–Galerkin and Petrov– Galerkin methods (Chapters 2, 3, and 5), the extremely accurate results by using the consistent version of Bresse–Timoshenko theory (Chapter 4 and Appendix J) and the Donnell shell model (Chapter 6). A comparison of our results with recent studies shows that the above methods constitute effective alternative techniques to exact solutions (which are given in Chapter 4) for studying the vibration and buckling properties of CNTs. The methods developed in this book are applicable to MWCNTs with an arbitrary number of walls, as well as for an individual CNT embedded in an elastic medium. Although in this study we do not deal with CNT’s embedded in elastic media, the dynamic characteristics of the latter are similar to those of single and multibeam models for SWCNT and MWCNT, respectively (see sections 1.2–1.4). Introduction 7

Books on nanotechnology include, among others, those by Brushan [BRU 04], Dresselhaus and Avouris [DRE 01a], Gdoutos [GDO 07], Goddard et al. [GOD 03], Gutkin and Ovid’ko [GUT 03], and Guz et al. [GUZ 10].

Studies of a general character are represented by papers by the following authors (listed in alphabetical order): Ball [BAL 01], Baughman et al.[BAU 02], Belin and Epron [BEL 05], Coleman et al. [COL 06], Endo et al. [END 08], Feynman [FEY 60], Guz and Rushchitsky [GUZ 03], Iijima and Ichihashi [IIJ 93], Lau and Hui [LAU 03], Liu et al. [LIU 06b], Majumdar [MAJ 01], Roukes [ROU 01a, ROU 01b], Zussman [ZUS 07], and others.

Mechanical properties of CNTs are studied by Gogolinskii et al. [GOG 04], Govindjee and Sackman [GOV 99], Hernandez et al. [HER 99], Ivanova and Morozov [IVA 05], Liew et al. [LIE 04], Kulik et al. [KUL 07], Parvaneh and Shariati [PAR 10], Salvetat et al. [SAL 06], Ruoff and Pugno [RUO 06b], Scarpa et al. [SCA 09], and others.

For references associated with nanocomposites, the reader may consult papers by Buryachenko and Roy [BUR 05], Daniel and Cho [Dan 10], Daniel et al. [DAN 03], Gdoutos and Daniel [GDO 09], Komarneni [KOM 92], Okotrub et al. [OKO 10], Thostenson et al. [THO 01], Wuite and Adali [WUI 05], Srivastava et al. [SRI 03], Formica et al. [FOR 10], Guz and Dekret [GUZ 09], Guz and Rushchitsky [GUZ 03], Li and Chou [LI 06], and others.

Finite-element methods in a nanotechnological setting were utilized by Fan et al. [FAN 09], Tserpes and Papanikos [TSE 05], Georgantzinos and Anifantis [GEO 09a], and others.

Experimental investigations on MWCNTs include works by Zhong et al. [ZHO 08a], Wei et al. [WEI 09a], Steele et al. [STE 10], Hüttel et al. [HÜT 09b], and others.

The state of the art in modeling and simulation of nanostrucutred materials and systems was given by Gates and Hinkley [GAT 03] and Liu et al. [LIU 06a], inter alia. 8 Carbon Nanotubes and Nanosensors

1.1. The need of determining the natural frequencies and buckling loads of CNTs

Natural frequencies are the intermediate step to find the response of the system. In other words, all objects have a set of natural frequencies at which they vibrate when excited. The actual natural frequency depends on the properties of the material that the object is made of and the dimensions of structure. The primary tools for vibrational analysis of beams are the Bernoulli–Euler beam equation or Bresse–Timoshenko beam equations that enable quick calculation of the natural frequencies of structural elements.

Leonhard Euler was also the first to calculate analytically what happens to a column when it is axially compressed. Initially the column remains straight as the compression increases up to the Euler’s critical buckling load. At the critical buckling load, the equilibrium transitions from stable to neutral. Any additional load will cause the column to buckle, and therefore, deflect laterally. The deformation on the CNTs disappears completely when the load is removed [WAN 07b].

1.2. Determination of natural frequencies of SWCNT as a uniform beam model and MWCNT during coaxial deflection

Although the dimensions of CNTs are from a few to tens of nanometers (nm) in diameter, and a few millimeters (mm) long, continuum models have been found to describe their mechanical behavior very well under many circumstances. The Bernoulli–Euler beam theory is used for small deformation and large aspect length to diameter ratios, Bresse–Timoshenko theory is used for small aspect ratios (see, for example, Chapter 5), and shell theory for larger and more complicated distortions (see, for example Chapter 6).

The governing differential equation of a uniform beam vibration was first written by Bernoulli–Euler around the year 1730:

∂4(),txy ∂2(),txy EI +ρA =0 [1.1] ∂x4 ∂t2 where E is the Young’s modulus of elasticity, I is the area moment of inertia of the cross section, ρ is the material density, A is the cross-sectional area, y(x, t) is the transverse displacement, x is the spatial coordinate along the Introduction 9 beam span, and t is the time. In the case of an SWCNT or MWCNT, the cross-section is a circular annulus of inner radius Rin and outer radius Rout, 22 44 and we have A =−π()RRout in and IRR=−π( out in ) 4. The SWCNT is considered as a single elastic beam, whereas all concentric tubes of a MWCNT remain coaxial during deflection and the interlayer radial displacement is ignored.

In order to determine the natural frequencies of a beam, equation [1.1] must be solved (see, for example [WEA 90]). Solution of equation [1.1] gives the natural frequency of the jth mode of vibration as:

EI = ()αω L 2 [1.2] j j ρAL4

where ()αLjis the root of an equation that depends on the boundary conditions, α a constant depending on the number j, and L is the beam length. In the case of a CNT clamped at one end and free at the other, the roots of ()αLjare obtained as follows:

()αLj=1.875; 4.694; 7.855; 10.996; … [1.3]

Experiments by Treacy et al. [TRE 96] and Poncharal et al. [PON 99], private communication with Prof. W.A. der Heer (October 30, 2008) and Wang et al. [WAN 01], show that the natural frequencies of CNTs can be evaluated by expression [1.2]. According to Poncharal et al. [PON 99], the experimental ratio of ω ω12 = 68.5 , which correlates well with the theoretical ratio ω ω12 = 2.6 of the clamped-free uniform beam. Also, they found that the position of the node in the ω2 is at 0.76L, which is very close to the theoretical value of approximately 0.8L.

1.3. Beam model of MWCNT

In this section, we show how MWCNTs are modeled by using the Bernoulli–Euler beam theory and in the following section how the analysis and results would change in case of CNTs embedded in an elastic medium. 10 Carbon Nanotubes and Nanosensors

Ru [RU 04] has given a description of a MWCNT: “In reality, individual tubes of a MWCNT could deform individually with nonzero interlayer radial displacements, while their individual deformations are coupled through the interlayer van der Waals interaction”. Yoon et al. [YOO 02], Ru [RU 04], Xu et al. [XU 06, XU 08b], and Natsuki et al. [NAT 08a] have shown that the free vibration of the MWCNT can be modeled by modifying the Bernoulli–Euler equation [1.1] to include van der Waals interaction forces and solving N coupled equations for an N-wall CNT of the form:

4422 1()2111ρ 11∂∂+∂∂=− twAxwEIwwc ()()44ρ 2∂∂+∂∂=−−− twAxwEIwwcwwc 2 23212122 22 [1.4] M 4422 −1()NNN −1EIwwc wNN xρAwNN ∂∂+∂∂=−− t

where x is the axial coordinate, t is the time, wxtj(,)is the transverse displacement of the jth tube, and Aj and Ij are its cross-sectional area and area moment of inertia, respectively. The subscripts j = 1, 2, …, N denote parameters of the innermost tube, its adjacent tube, outermost tube, etc. The coefficient cj denotes the van der Waals interlayer interaction coefficient between the jth and (j + 1)th nanotubes (j = 1, 2, …, N–1).

The fundamental frequencies of a MWCNT under various boundary conditions and various length-to-diameter ratios are in the terahertz (THz) range. This topic is discussed extensively in Chapters 2–4.

1.4. CNTs embedded in an elastic medium

For MWCNTs embedded in an elastic medium such as a polymer matrix, the effect of the matrix on the fundamental natural frequencies of the MWCNT has been estimated by using a beam on elastic foundation model by Yoon et al. [YOO 03]. In this model, expressions of equation [1.4] are modified by adding a distributed elastic reaction force (Winkler-type elastic reaction force per unit length) to the last expression of equation [1.4], which applies to the outer tube in a MWNT as: Introduction 11

4 4 2 2 ()121 1 1 ρ 1 1 ∂∂+∂∂=− twAxwEIwwc ()()4 4 ρ 2 ∂∂+∂∂=−−− twAxwEIwwcwwc 2 121232 2 2 2 2 [1.5] M 4 4 2 2 ()−1 (NNN −1 ) EIwwcxp NN ρ NN ∂∂+∂∂=−− twAxw

where ()−= kwxp N ()x, N ()xw is the deflection of the embedded CNT, and k is a constant that depends on the material in which the MWCNT is embedded. Ru [RU 04] defines k as a “spring constant of the surrounding elastic medium which may depend only on the Young’s modulus of the surrounding elastic medium and the outermost diameter of the CNT, but also on the wave-length of the deformed CNT. The minus sign on the right-hand side of p(x) indicates that the interaction pressure is opposite to the deflection”.

Also, the elastic foundation treatment is used for the shell models by several researchers. The reader is referred to review papers of Yoon et al. [YOO 03], Ru [RU 01, RU 04], Parnes and Cheskis [PAR 02], Lourie et al. [LOU 98], and Thostenson and Chou [THO 04], for details on this interesting topic.

In the case of a SWCNT embedded in an elastic medium, the jth order natural frequency in equation [1.2] (see [RU 04] and [YOO 03]) becomes:

()α 4+ kEIL [1.6] ω = j . j ρAL4

The fundamental frequency of a SWCNT under various boundary conditions is in the gigahertz (GHz) range, and when the nanotube is embedded in a polymer matrix that provides elastic restraint, the respective frequencies are increased.

In the case of CNTs-based reinforcement applications, there are many outstanding problems that must be overcome. First, the properties of the individual CNTs must be optimized. Second, the CNTs must be efficiently bonded to the materials they are reinforcing so that they actually carry the 12 Carbon Nanotubes and Nanosensors loads. Third, the load must be distributed within the nanotube itself to ensure that the outermost layer does not shear off. Determination of effective moduli of the embedded CNTs is another area of future research. The following chapters discuss effective techniques for studying the vibration, buckling, and impact behavior of CNTs, along with the study of carbon nanosensors. Chapter 2

Fundamental Natural Frequencies of Double-Walled Carbon Nanotubes

This chapter deals with the evaluation of fundamental natural frequencies of double-walled carbon nanotubes (DWCNTs) under various boundary conditions. The Bubnov–Galerkin and Petrov–Galerkin methods are applied to derive the expressions for natural frequencies. This is the first time that explicit expressions obtained for the natural frequencies are reported. These could be useful to a designer to estimate the fundamental frequency in each of two series that the DWCNT possess.

2.1. Background

Vibrations of DWCNTs have been considered in several papers. Yoon et al. [YOO 02], Xu et al. [XU 06, XU 08b], and Ru [RU 04] studied the free vibrations of a DWCNT that is composed of two coaxial single-walled carbon nanotubes (SWCNTs) interacting with each other by the interlayer van der Waals forces. Therefore, the inner and the outer carbon nanotubes (CNTs) are modeled as two individual elastic beams (see [YOO 02], [XU 06, XU 08b], and [RU 04]). In these studies, the Bernoulli–Euler beam model has been used to derive exact expressions for the natural frequencies under various boundary conditions. The results showed that DWCNTs have frequencies in the range of terahertz (THz). 14 Carbon Nanotubes and Nanosensors

Also, in the study of vibration of CNTs the Timoshenko beam model has been used for short length-to-diameter ratios, which allow for the effect of transverse shear deformation [YOO 05, WAN 06a]. Likewise, the shell models have been recently applied by He et al. [HE 06], Ru [RU 01], and Wang et al. [WAN 07a].

Ru [RU 04] stresses that “carbon MWCNTs are different from traditional elastic beams due to their hollow multilayer structure and the associated interlayer van der Waals forces”. The exact solutions lead to the need of solving transcendental equations. Our work demonstrates that they can be usefully supplemented by simple solutions.

Vibrations of CNTs are studied in numerous papers. These papers include those by Aydoğdu [AYD 08a], Babic [BAB 03], Beidermann et al. [BEI 09], Dalir et al. [DAL 09], Dong and Wang [DON 07], Ece and Aydoğdu [ECE 07], Eklund et al. [EKL 95], Kuang and Leung [KUA 08], Li and Chou [LI 04a], Eremeyev et al. [ERE 06, ERE 07, ERE 08], Garcia-Sanchez et al. [GAR 07], Hüttel [HÜT 10a], Li et al. [LI 09a], Mitra and Gopalukrishnan [MIT 07], Natsuki et al. [NAT 05], Peng [PEN 06], Popov and Lambin [POP 06b], Popov et al. [POP 09], Rivera et al. [RIV 03], Sun and Liu [SUN 07], Talebian et al. [TAL 10], Vedernikov and Chaplik [VED 04], Wang and Dai [WAN 06h], Yang et al. [YAN 09b], Zhao et al. [ZHA 03], Zheng and Jiang [ZHE 02a, 02b], Hüttel et al. [HÜT 09a, HÜT 09b], Witkamp et al. [WIT 06], Maninder and Gibson [MAN 09], Mir et al. [MIR 08], Brovko and Tunguskova [BRO 09], Chowdhury et al. [CHO 10a], Dikande [DIK 04], Amlani et al. [AML 09], Benoit et al. [BEN 06], Buisson et al. [BUI 03], Elishakoff and Pentaras [ELI 9a, ELI 09c], Elishakoff et al. [ELI 11a], Farshi et al. [FAR 10], Georgantzinos and Anifantis [GEO 09a], Ghorbanpour et al. [GHO 10], Gibson et al. [GIB 07], Gupta et al. [GUP 09, GUP 10], Hailiang et al. [HAI 08], Karaoglu and Aydoğdu [KAR 10], Kienle and Léonard [KIE 00], Li and Chou [LI 04b], and others.

Vibrations of CNTs in the context of non-local elasticity are treated by Adali [ADA 09a, ADA 9b], Ece and Aydoğdu [ECE 07], Kiani [KIA 10a, KIA 10b), Ke et al. [KE 09], Kucuk et al. [KUC 10], Li and Kardomateas [LI 07b], Zhang et al. [ZHA 10a], Lee and Chang [LEE 09b], and others.

In this chapter, approximate solutions are found by using Bubnov–Galerkin ([BUB 13] and [GAL 15]) and Petrov–Galerkin [PET 40] methods. Explicit formulas of natural frequencies are derived for DWCNTs