Hard-to-Soft Transition in Radial Buckling of Multi-Concentric Nanocylinders


We investigate the cross-sectional buckling of multi-concentric tubular nanomaterials, which are called multiwalled carbon nanotubes (MWNTs), using an analysis based on thin-shell theory. MWNTs under hydrostatic pressure experience radial buckling. As a result of this, different buckling modes are obtained depending on the inter-tube separation d as well as the number of constituent tubes N and the innermost tube diameter. All of the buckling modes are classified into two deformation phases. In the first phase, which corresponds to an elliptic deformation, the radial stiffness increases rapidly with increasing N. In contrast, the second phase yields wavy, corrugated structures along the circumference for which the radial stiffness declines with increasing N. The hard-to-soft phase transition in radial buckling is a direct consequence of the core-shell structure of MWNTs. Special attention is devoted to how the variation in d affects the critical tube number Nc, which separates the two deformation phases observed in N -walled nanotubes, i.e., the elliptic phase for N < Nc and the corrugated phase for N > Nc. We demonstrate that a larger d tends to result in a smaller Nc, which is attributed to the primary role of the interatomic forces between concentric tubes in the hard-to-soft transition during the radial buckling of MWNTs.

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S. Park, M. Sato, T. Ikeda and H. Shima, "Hard-to-Soft Transition in Radial Buckling of Multi-Concentric Nanocylinders," World Journal of Mechanics, Vol. 2 No. 1, 2012, pp. 42-50. doi: 10.4236/wjm.2012.21006.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] D. O. Brush and B. O. Almroth, “Buckling of Bars, Plates, and Shells,” McGraw-Hill, New York, 1975.
[2] M. Sato and M. H. Patel, “Exact and Simplified Estimations for Elastic Buckling Pressures of Structural Pipe-in-Pipe Cross Sections under External Hydrostatic Pressure,” Journal of Marine Science and Technology, Vol. 12, No. 4, 2007, pp. 251-262. doi:10.1007/s00773-007-0244-y
[3] M. Sato, M. H. Patel and F. Trarieux, “Static Displacement and Elastic Buckling Characteristics of Structural Pipe-in-Pipe Cross-Sections,” Structural Engineering and Mechanics, Vol. 30, 2008, pp. 263-278.
[4] H. Shima, “Buckling of Carbon Nanotubes: A State of the Art Review,” Materials, Vol. 5, No. 1, 2012, pp. 47-84. doi:10.3390/ma5010047
[5] R. Zhang, Q. Wen, W. Qian, D. Sheng, Q. Zhang and F. Wei, “Superstrong Ultralong Carbon Nanotubes for Mechanical Energy Storage,” Advanced Materials, Vol. 23, No. 30, 2011, pp. 3387-3391. doi:10.1002/adma.201100344
[6] B. I. Yakobson, C. J. Brabec and J. Bernholc, “Nanomechanics of Carbon Tubes: Instabilities beyond Linear Response,” Physical Review Letters, Vol. 76, No. 14, 1996, pp. 2511-2514. doi:10.1103/PhysRevLett.76.2511
[7] C. Q. Ru, “Axially Compressed Buckling of a Doublewalled Carbon Nanotube Embedded in an Elastic Medium,” Journal of the Mechanics and Physics of Solids, Vol. 49, No. 6, 2001, pp. 1265-1279. doi:10.1016/S0022-5096(00)00079-X
[8] B. Ni, S. B. Sinnott, P. T. Mikulski and J. A. Harrison, “Compression of Carbon Nanotubes Filled with C60, CH4, or Ne: Predictions from Molecular Dynamics Simulations,” Physical Review Letters, Vol. 88, 2002, pp. 205505: 1-205505:4. doi:10.1103/PhysRevLett.88.205505
[9] M. J. Buehler, J. Kong and H. J. Gao, “Deformation Mechanism of Very Long Single-Wall Carbon Nanotubes Subject to Compressive Loading,” Journal of Engineering Materials and Technology, Vol. 126, No. 3, 2004, pp. 245-249. doi:10.1115/1.1751181
[10] A. Pantano, M. C. Boyce and D. M. Parks, “Mechanics of Axial Compression of Single- and Multi-Wall Carbon Nanotubes,” Journal of Engineering Materials and Tech- nology, Vol. 126, No. 3, 2004, pp. 279-284. doi:10.1115/1.1752926
[11] J. Tang, J. C. Qin, T. Sasaki, M. Yudasaka, A. Matsushita and S. Iijima, “Compressibility and Polygonization of Single-Walled Carbon Nanotubes under Hydrostatic Pre- ssure,” Physical Review Letters, Vol. 85, No. 9, 2000, pp. 1887-1889. doi:10.1103/PhysRevLett.85.1887
[12] A. Pantano, D. M. Parks and M. C. Boyce, “Mechanics of Deformation of Single- and Multi-Wall Carbon Nanotubes,” Journal of the Mechanics and Physics of Solids, Vol. 52, No. 4, 2004, pp. 789-821. doi:10.1016/j.jmps.2003.08.004
[13] J. A. Elliott, L. K. W. Sandler, A. H. Windle, R. J. Young and M. S. P. Shaffer, “Collapse of Single-Wall Carbon Nanotubes Is Diameter Dependent,” Physical Review Le- tters, Vol. 92, 2004, pp. 095501:1-095501:4. doi:10.1103/PhysRevLett.92.095501
[14] H. Shima and M. Sato, “Multiple Radial Corrugations in Multiwall Carbon Nanotubes under Pressure,” Nanotechnology, Vol. 19, 2008, pp. 495705:1-495705:8. doi:10.1088/0957-4484/19/49/495705
[15] J. Peng, J. Wu, K. C. Hwang, J. Song and Y. Huang, “Can a Single-Wall Carbon Nanotube Be Modeled as a Thin Shell?” Journal of the Mechanics and Physics of Solids, Vol. 56, No. 6, 2008, pp. 2213-2224. doi:10.1016/j.jmps.2008.01.004
[16] H. Shima and M. Sato, “Pressure-Induced Structural Transitions in Multi-Walled Carbon Nanotubes,” Physica Status Solidi (a), Vol. 206, 2009, pp. 2228-2233. doi:10.1002/pssa.200881706
[17] M. Sato and H.Shima, “Buckling Characteristics of Multiwalled Carbon Nanotubes under External Pressure,” Interaction and Multiscale Mechanics: An International Journal, Vol. 2, 2009, pp. 209-222.
[18] A. P. M. Barboza, H. Chacham and B. R. A. Neves, “Universal Response of Single-Wall Carbon Nanotubes to Radial Compression,” Physical Review Letters, Vol. 102, 2009, pp. 025501:1-025501:4. doi:10.1103/PhysRevLett.102.025501
[19] H. Shima, M. Sato, K. Iiboshi, S. Ghosh and M. Arroyo, “Diverse Corrugation Pattern in Radially Shrinking Carbon Nanotubes,” Physical Review B, Vol. 82, 2010, pp. 085401:1-085401:7. doi:10.1103/PhysRevB.82.085401
[20] M. Sato, H. Shima and K. Iiboshi, “Core-Tube Morphology of Multiwall Carbon Nanotubes,” International Jour- nal of Modern Physics B, Vol. 24, No. 1-2, 2010, pp. 288- 294. doi:10.1142/S0217979210064228
[21] X. Huang, W. Liang and S. Zhang, “Radial Corrugations of Multi-Walled Carbon Nanotubes Driven by Inter-Wall Nonbonding Interactions,” Nanoscale Research Letters, Vol. 6, 2011, pp. 53-58. doi:10.1007/s11671-010-9801-0
[22] H. Shima, S. Ghosh, M. Arroyo, K. Iiboshi and M. Sato, “Thin-Shell Theory Based Analysis of Radially Pressurized Multiwall Carbon Nanotubes,” Computational Materials Science, Vol. 52, No. 1, 2012, pp. 90-94. doi:10.1016/j.commatsci.2011.04.005
[23] S. Iijima, C. Brabec, A. Maiti and J. Bernholc, “Structural Flexibility of Carbon Nanotubes,” Journal of Chemical Physics, Vol. 104, No. 5, 1996, pp. 2089-2092. doi:10.1063/1.470966
[24] M. R. Falvo, G. J. Clary, R. M. Taylor II, V. Chi, F. P. Brooks Jr., S. Washburn and R. Superfine, “Bending and Buckling of Carbon Nanotubes under Large Strain,” Nature, Vol. 389, 1997, pp. 582-584. doi:10.1038/39282
[25] P. Poncharal, Z. L. Wang, D. Ugarte and W. A. de Heer, “Electrostatic Deflections and Electromechanical Resonances of Carbon Nanotubes,” Science, Vol. 283, No. 5407, 1999, pp. 1513-1516. doi:10.1126/science.283.5407.1513
[26] Y. Shibutani and S. Ogata, “Mechanical Integrity of Carbon Nanotubes for Bending and Torsion,” Modelling and Simulation in Materials Science and Engineering, Vol. 12, No. 4, 2004, pp. 599-610. doi:10.1088/0965-0393/12/4/003
[27] A. Kutana and K. P. Giapis, “Transient Deformation Regime in Bending of Single-Walled Carbon Nanotubes,” Physical Review Letters, Vol. 97, 2006, pp. pp.245501: 1-245501:4. doi:10.1103/PhysRevLett.97.245501
[28] H. K. Yang and X. Wang, “Bending Stability of Multi-Wall Carbon Nanotubes Embedded in an Elastic Medium,” Modelling and Simulation in Materials Science and Engineering, Vol. 14, No. 1, 2006, pp. 99-116. doi:10.1088/0965-0393/14/1/008
[29] I. Arias and M. Arroyo, “Size-Dependent Nonlinear Elastic Scaling of Multiwalled Carbon Nanotubes,” Physical Review Letters, Vol. 100, 2008, pp. 085503:1-085503:4. doi:10.1103/PhysRevLett.100.085503
[30] Q. Wang, “Torsional Buckling of Double-Walled Carbon Nanotubes,” Carbon, Vol. 46, No. 8, 2008, pp. 1172- 1174. doi:10.1016/j.carbon.2008.03.025
[31] M. Arroyo and I. Arias, “Rippling and a Phase-Trans- forming Mesoscopic Model for Multiwalled Carbon Na- notubes,” Journal of the Mechanics and Physics of Solids, Vol. 56, No. 4, 2008, pp. 1224-1244. doi:10.1016/j.jmps.2007.10.001
[32] B. W. Jeong and S. B. Sinnott, “Unique Buckling Responses of Multi-Walled Carbon Nanotubes Incorporated as Torsion Springs,” Carbon, Vol. 48, No. 6, 2010, pp. 1697-1701. doi:10.1016/j.carbon.2009.12.048
[33] H. Shima and M. Sato, “Elastic and Plastic Deformation of Carbon Nanotoubes,” Pan Stanford Publishing, Singapore, 2012.
[34] R. Saito, M. S. Dresselhaus and G. Dresselhaus, “Physical Properties of Carbon Nanotubes,” World Scientific Publishing Company, 1998.
[35] A. Loiseau, P. Launois. P. Petit, S. Roche and J. -P. Salvetat, “Understanding Carbon Nanotubes: From Basics to Application,” Springer-Verlag, Berlin, 2006.
[36] C. Q. Ru, “Column Buckling of Multiwalled Carbon Na- notubes with Interlayer Radial Displacements,” Physical Review B, Vol. 62, 2000, pp. 16962-16967. doi:10.1103/PhysRevB.62.16962
[37] C. Y. Wang, C. Q. Ru and A. Mioduchowski, “Axially Compressed Buckling of Pressured Multiwall Carbon Nanotubes,” International Journal of Solids and Structures, Vol. 40, No. 15, 2003, pp. 3893-3911. doi:10.1016/S0020-7683(03)00213-0
[38] H. S. Shen, “Postbuckling Prediction of Double-Walled Carbon Nanotubes under Hydrostatic Pressure,” International Journal of Solids and Structures, Vol. 41, No. 9-10, 2004, pp. 2643-2657. doi:10.1016/j.ijsolstr.2003.11.028
[39] X. Q. He, S. Kitipornchai and K. M. Liew, “Buckling Analysis of Multi-Walled Carbon Nanotubes: A Continuum Model Accounting for van der Waals Interaction,” Journal of the Mechanics and Physics of Solids, Vol. 53, No. 2, 2005, pp. 303-326. doi:10.1016/j.jmps.2004.08.003
[40] N. Silvestre, “Length Dependence of Critical Measures in Single-Walled Carbon Nanotubes,” International Journal of Solids and Structures, Vol. 45, No. 18-19, 2008, pp. 4902-4920. doi:10.1016/j.ijsolstr.2008.04.029
[41] N. Silvestre, C. M. Wang, Y. Y. Zhang and Y. Xiang, “Sanders Shell Model for Buckling of Single-Walled Carbon Nanotubes with Small Aspect Ratio,” Composite Structures, Vol. 93, No. 7, 2011, pp. 1683-1691. doi:10.1016/j.compstruct.2011.01.004
[42] S. S. Gupta, F. G. Bosco and R. C. Batra, “Wall Thickness and Elastic Moduli of Single-Walled Carbon Nanotubes from Frequencies of Axial, Torsional and Inextensional Modes of Vibration,” Computational Materials Science, Vol. 47, 2010, pp. 1049-1059. doi:10.1016/j.commatsci.2009.12.007
[43] K. N. Kudin, G. E. Scuseria and B. I. Yakobson, “C2F, BN, and C Nanoshell Elasticity from ab initio Computations,” Physical Review B, Vol. 64, 2001, pp. 235406: 1-235406:10. doi:10.1103/PhysRevB.64.235406
[44] W. B. Lu, B. Liu, J. Wu, J. Xiao, K. C. Hwang, S. Y. Fu and Y. Huang, “Continuum Modeling of van der Waals Interactions between Carbon Nanotube Walls,” Applied Physics Letters, Vol. 94, 2009, pp. 101917:1-101917:3. doi:10.1063/1.3099023
[45] L. A. Girifalco, M. Hodak and R. S. Lee, “Carbon Nanotubes, Buckyballs, Ropes, and a Universal Graphitic Potential,” Physical Review B, Vol. 62, No. 19, 2000, pp. 13104-13110. doi:10.1103/PhysRevB.62.13104
[46] K. Koziol, M. Shaffer and A. Windle, “Three-Dimensional Internal Order in Multiwalled Carbon Nanotubes Grown by Chemical Vapor Deposition,” Advanced Materials, Vol. 17, 2005, pp. 760-763. doi:10.1002/adma.200401791
[47] C. Ducati, K. Koziol, S. Friedrichs, T. J. V. Yates, M. S. Shaffer, P. A. Midgley and A. H. Windle, “Crystallographic Order in Multi-Walled Carbon Nanotubes Synthesized in the Presence of Nitrogen,” Small, Vol. 2, No. 6, 2006, pp. 774-784. doi:10.1002/smll.200500513

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