COMPREHENSIVE MOLECULAR DYNAMICS SIMULATIONS OF CARBON NANOTUBES UNDER AXIAL FORCE OR TORSION OR VIBRATION AND NEW CONTINUUM MODELS AMAR NATH ROY CHOWDHURY (B.Eng. (Hons.), Jadavpur University. M.Tech., Indian Institute of Technology Bombay, India) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2014 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. ______________________________________ Amar Nath Roy Chowdhury December 2014 iii ACKNOWLEDGMENTS I would like to express my sincere gratitude towards Professor Wang Chien Ming for giving me the opportunity to pursue a doctoral research in the Department of Civil and Environmental Engineering at National University of Singapore. Occasionally, during my PhD study, I felt disappointed and demotivated by getting stuck on research problems, but thanks to Prof Wang’s valuable guidance, I was able to regain hope to carry out the research work. I would also like to thank Dr. Adrian Koh of the Department of Mechanical Engineering for helping me to enrich my analytical and critical thinking capabilities. I want to express special thanks to both of them for listening to all of my ideas, and for mentoring me to complete the PhD research. Because of Prof. Wang and Dr. Koh’s patience and expert advices, I am able to sculpt my research work in the form of this thesis within the desired period of time. In the first semester, I was also lucky to get the opportunity to interact with Dr. Prakash Thamburaja who was a former faculty the Department of Mechanical Engineering at NUS. He helped me to understand some advanced concepts of continuum mechanics that were and are useful for my research. Thanks to Dr. Yingyan Zhang of University of Western Sydney for helping me to learn molecular dynamics techniques using LAMMPS. Also, I wish to acknowledge Dr. Zhi Yung Tay of University of Edinburgh for introducing me to ABAQUS. Special thanks go to NUS-HPC for the terrific computational resources and to NUS library for the vast source of literature, without them I would have not been able to finish the work within four years. v I am deeply thankful to my uncle Deb Kumar Mitra, my father Biswanath Roy Chowdhury, two of my aunts Lina Mitra and Bharati Sengupta, and my mother Mina Roy Chowdhury for being so awesome, caring and supporting. My special thanks also go to three of my very close childhood friends Sandip Saha, Sandip Dutta and Sumit Mukherjee who were always with me to overcome hard times in the last four years, thereby assisting me to progress in my academic career. I also want to thank my friends and four-years-roommates Nirmalya Bag, and Shubham Duttagupta for making the PhD life fun-filled. Last but not the least I want to dedicate this thesis to the loving memories of my uncle Ashoke Sengupta whose feats and success stories motivated me to pursue scientific career and also to my beloved aunt Shila Mitra. vi CONTENTS Declaration iii Acknowledgments v Contents vii Extended summary xiii List of tables xvii List of figures . xix List of symbols and acronyms .xxvii Chapter Introduction 1.1 Properties and applications of carbon nanotubes 1.1.1 Geometric properties of carbon nanotubes 1.1.2 Mechanical properties of CNT and its characterization 1.1.3 Applications of CNT 12 1.2 Computational models to study mechanics of carbon nanotubes . 15 1.2.1 Atomistic models of carbon nanotubes 16 1.2.2 Continuum models of CNT 19 1.3 Literature review . 22 1.3.1 Atomistic simulations of compression and torsional buckling of CNT . 22 1.3.2 Atomistic and continuum models for CNT under tension . 31 1.3.3 Vibration frequencies of CNT using atomistic calculations 33 1.4 Objectives of thesis . 35 vii 1.5 Layout of thesis . 37 Chapter Details on MD simulation of CNT 39 2.1 Mathematical construct of CNT 39 2.2 Description of simulation steps . 41 2.2.1 Interatomic potential 43 2.2.2 Energy relaxation . 49 2.2.3 Integration time-step 52 2.2.4 Incremental displacement or displacement rate . 53 2.2.5 Relaxation time Ts 54 2.2.6 Thermostat . 59 2.2.7 Barostat 62 2.2.8 Brief description of MD simulation steps with NVT ensemble 63 2.3 Comparison of MD simulations with NPT and NVT ensemble . 63 2.4 Summary and key findings 65 Chapter MD simulations of CNT under compression . 67 3.1 Load deformation behaviour of CNT under compression 67 3.1.1 Displacement rate for MD simulation 67 3.1.2 Stress-strain response . 69 3.2 Buckling of CNT under compression 72 3.2.1 Definition buckling 73 3.2.2 Compressive buckling results for non-chiral SWCNTs . 75 viii 3.2.3 Effect of θ on buckling properties . 78 3.2.4 Effect of wall numbers on buckling characteristics of CNT 81 3.3 Summary and Conclusions 83 Chapter MD simulations of CNT under torsion 85 4.1 Torque-twist response of SWCNT 85 4.1.1 Twisting rate for MD simulation . 85 4.1.2 Shear stress-shear strain response 87 4.2 Torsional buckling of CNT . 96 4.2.1 Definition of torsional buckling . 96 4.2.2 Torsional buckling results for non-chiral SWCNTs 98 4.2.3 Effect of θ and D on torsional buckling of SWCNT . 100 4.2.4 Effect of wall numbers on torsional buckling of CNT 102 4.3 Summary and Conclusions 103 Chapter Thick shell model for CNT 105 5.1 Description of cylindrical shell theory 105 5.2 Compressive buckling of CNT 108 5.2.1 Calibration of E of SWCNTs . 109 5.2.2 Inter-tube van der Waals interaction in MWCNT . 112 5.2.3 Finite element model for buckling analysis of CNTs 113 5.2.4 Compressive buckling of CNT: comparison of thick shell and MD results 115 5.3 Torsional buckling of CNT . 125 ix 5.3.1 Thick shell results for torsional buckling of armchair SWCNTs 126 5.3.2 Torsional buckling of chiral SWCNTs 130 5.3.3 Torsional buckling of armchair and zigzag MWCNTs 135 5.4 Summary and Conclusions 139 Chapter Continuum models for CNT under tension . 141 6.1 Introduction . 141 6.2 Atomistic simulations 143 6.3 Membrane-shell model of SWCNT 147 6.3.1 Kinematics of axial deformation 148 6.3.2 Constitutive relation . 149 6.3.3 Calibration of material parameters . 152 6.3.4 Advantages and disadvantages of membrane-shell model 155 6.4 Hyper-elastic continuum model of CNT with softening . 155 6.4.1 Softening hyper-elasticity 156 6.4.2 Thermal effect 161 6.4.3 Effect of hydrostatic pressure 163 6.4.4 Advantages and disadvantages of hyper-elastic continuum model . 167 6.5 Summary and Conclusions 168 Chapter Modal analysis of CNT 171 7.1 Calculation of vibration frequency from MD simulation 171 7.2 Operational modal analysis using TDD 172 x nested system of cylindrical Shells." 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"Nonlinearelastic membrane-shell model for single-walled carbon nanotubes under uni-axial deformation." Computational Materials Science 97, pp. 237-244.  Roy Chowdhury, A. N., Wang, C. M., and Koh, S. J. A. (2014b). "Continuum shell model for buckling of armchair carbon nanotubes under compression or torsion." International Journal of Applied Mechanics 6, pp. 1-25. 217 Book chapter  Wang, C. M., Roy Chowdhury, A. N, Koh, S. J. A., and Zhang, Y. Y. (2014). Molecular dynamics simulation and continuum shell model for buckling analysis of carbon nanotubes. Modeling of carbon nanotubes, graphene and their composites, K. I. Tserpes, and N. Silvestre, eds., Springer International Publishing, pp. 239-273. Conference articles  Roy Chowdhury, A. N., Koh, S. J. A., and Wang, C. M., Nonlinear-elastic membrane-shell model for single walled carbon nanotubes. 23rd International Workshop on Computational Mechanics of Materials, Singapore, October 2-4, 2013. (paper presented)  Roy Chowdhury, A. N., and Wang, C. M., Molecular simulation results for axial buckling of double walled carbon nanotubes, The 7th International Conference on Advances in Steel Structures, Nanjing, China, April 14-16, 2012 (paper presented).  Wang, C. M., and Roy Chowdhury, A. N., Molecular dynamics simulation results for buckling of carbon nanotubes with small aspect ratios. International Congress on Computational Mechanics and Simulation (ICCMS), IIT Hyderabad, December 2012 (Keynote Lecture). 218 [...]... Existing molecular dynamics (MD) simulation results for CNT under uni -axial deformation, torsion, and vibration are not comprehensive Moreover, in many cases discrepancies are observed in MD simulation results reported by different researchers For that purpose, extensive classical MD simulations are performed to generate accurate benchmark results for CNT under uni -axial deformation, torsion and vibration. .. Discrepancies in reported cr of SWCNT(5,5) 27 Figure 1.11: Discrepancies in reported cr of DWCNT((5,5),(10,10)) 28 Figure 2.1: Hexagonal graphene lattice and CNT unit cell 40 Figure 2.2: Molecular dynamics simulation set-up for CNT under uniaxial deformation /torsion 41 Figure 2.3: Molecular dynamics simulation set-up for CNT under uniaxial deformation /torsion using NPT ensemble... L, D, wall number, and θ of non-chiral SWCNTs For chiral CNTs, Mcr and cr depend on twist direction also Mcr and cr of chiral SWCNT under anti-clockwise torsion, is greater than the Mcr and cr of chiral SWCNT under clockwise torsion Although in case of MWCNT the τ- relation is not affected by number of walls, but the torsional buckling characteristics depend on wall number For MWCNT, Mcr increases... Summary and Conclusions 187 8.2 Future Studies 192 References 197 List of author’s publications 217 xi EXTENDED SUMMARY The main objective of this thesis is to obtain continuum models suitable to analyze mechanics of carbon nanotube (CNT) under uni -axial deformation, torsion, and vibration In general, continuum models of CNT are calibrated from atomistic simulations. .. L, and θ In case of chiral SWCNT, the τ- response also depends on twist direction because the carbon- carbon bonds are asymmetrically arranged along the length and perimeter of SWCNT, and the force deformation relation of carbon- carbon bond is different under compression and tension The MD simulation results reveal that the shear modulus (G) of non-chiral SWCNT depends on D As D increases, the G of. .. buckling results for zigzag and chiral DWCNTs with L/Di ≤ 10 under axial load 82 Table 4.1: MD buckling results for armchair SWCNTs with L/D ≤ 10 under torsion9 8 Table 4.2: MD buckling results for zigzag SWCNTs with L/D ≤ 10 under torsion 99 Table 4.3: MD buckling results for armchair DWCNTs with L/Di ≤ 10 under torsion 102 Table 4.4: MD buckling results for zigzag DWCNTs... GPa The G of chiral SWCNT also depends on twist direction SWCNT with chiral angle 15.5o has the highest G under clockwise torsion, but G of the same SWCNT is the lowest under anticlockwise torsion Occurrence of torsional buckling is indicated by the degradation of the slope of (torque) MZ versus (end rotation)  curve The critical buckling torque (Mcr) and critical buckling end-rotation (cr) of CNT depend... shell theory are very close to MD simulation results xvi LIST OF TABLES Table 1.1 E of CNT from various experiments 5 Table 1.2 Mechanical properties of various fiber materials 11 Table 1.3: Electrical and thermal properties of CNT 12 Table 1.4: Summary of SWCNT buckling under torsion 29 Table 1.5: Summary of DWCNT buckling under torsion 31 Table 2.1: Parameters for original... calculate the vibration frequencies and mode shapes of SWCNTs from MD simulation trajectories Comparing the vibration frequencies of various chiral SWCNTs, we find that the vibration frequencies of SWCNT are independent of θ, but similar to a thick cylindrical shell, the vibration frequencies depend on D and L of SWCNT Thick shell theory is used to calculate vibration frequencies We observe that the vibration. .. interaction for carbon- carbon bond 45 Table 2.2: Influence of Ts on critical buckling load 55 Table 3.1: MD buckling results for armchair SWCNTs under axial load 75 Table 3.2: MD buckling results for zigzag SWCNTs under axial load 76 Table 3.3: Effect of chiral indices on MD buckling results for SWCNTs 78 Table 3.4: MD buckling results for armchair DWCNTs with L/Di ≤ 10 under axial load . COMPREHENSIVE MOLECULAR DYNAMICS SIMULATIONS OF CARBON NANOTUBES UNDER AXIAL FORCE OR TORSION OR VIBRATION AND NEW CONTINUUM MODELS AMAR NATH. objective of this thesis is to obtain continuum models suitable to analyze mechanics of carbon nanotube (CNT) under uni -axial deformation, torsion, and vibration. In general, continuum models of CNT. models of carbon nanotubes 16 1.2.2 Continuum models of CNT 19 1.3 Literature review 22 1.3.1 Atomistic simulations of compression and torsional buckling of CNT . 22 1.3.2 Atomistic and continuum