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Effects of layer number and initial pressure on continuum-based buckling analysis of multi-walled carbon nanotubes accounting for van der Waals interaction

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Abstract

The structural instability of multi-walled carbon nanotubes (MWCNTs) has captured extensive attention due to the unique characteristic of extremely thin hollow cylinder structure. The previous studies usually focus on the buckling behavior without considering the effects of the wall number and initial pressure. In this paper, the axial buckling behavior of MWCNTs with the length-to-outermost radius ratio less than 20 is investigated within the framework of the Donnell shell theory. The governing equations for the infinitesimal buckling of MWCNTs are established, accounting for the van der Waals (vdW) interaction between layers. The effects of the wall number, initial pressure prior to buckling, and aspect ratio on the critical buckling mode, buckling load, and buckling strain are discussed, respectively. Specially, the four-walled and twenty-walled CNTs are studied in detail, indicating the fact that the buckling instability may occur in other layers besides the outermost layer. The obtained results extend the buckling analysis of the continuum-based model, and provide theoretical support for the application of CNTs.

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References

  1. KUMAR, S., RANI, R., DILBAGHI, N., TANKESHWAR, K., and KIM, K. H. Carbon nanotubes: a novel material for multifaceted applications in human healthcare. Chemical Society Reviews, 46, 158–196 (2017)

    Article  Google Scholar 

  2. YUAN, Y., ZHAO, K., SAHMANI, S., and SAFAEI, B. Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes. Applied Mathematics and Mechanics (English Edition), 41(4), 587–604 (2020) https://doi.org/10.1007/s10483-020-2600-6

    Article  MathSciNet  MATH  Google Scholar 

  3. SUN, T., GUO, J., and PAN, E. Nonlocal vibration and buckling of two-dimensional layered quasicrystal nanoplates embedded in an elastic medium. Applied Mathematics and Mechanics (English Edition), 42(8), 1077–1094 (2021) https://doi.org/10.1007/s10483-021-2743-6

    Article  MathSciNet  MATH  Google Scholar 

  4. IIJIMA, S. Helical microtubes of graphitic carbon. nature, 354, 56–58 (1991)

    Article  Google Scholar 

  5. BAI, Y., ZHANG, R., YE, X., ZHU, Z., XIE, H., SHEN, B., CAI, D., LIU, B., ZHANG, C., JIA, Z., ZHANG, S., LI, X., and WEI, F. Carbon nanotube bundles with tensile strength over 80GPa. Nature Nanotechnology, 13, 589–595 (2018)

    Article  Google Scholar 

  6. BHATTACHARYYA, A., SETH, G. S., KUMAR, R., and CHAMKHA, A. J. Simulation of Cattaneo-Christov heat flux on the flow of single and multi-walled carbon nanotubes between two stretchable coaxial rotating disks. Journal of Thermal Analysis and Calorimetry, 139, 1655–1670 (2019)

    Article  Google Scholar 

  7. ABBASI, S. A., KIM, T. H., SOMU, S., WANG, H., CHAI, Z., UPMANYU, M., and BUSNAINA, A. Fabrication of a nanoelectromechanical bistable switch using directed assembly of SWCNTs. Journal of Physics D: Applied Physics, 53, 23LT02 (2020)

    Article  Google Scholar 

  8. SAMY, M. M., MOHAMED, M. G., EL-MAHDY, A. F. M., MANSOURE, T. H., WU, K. C., and KUO, S. W. High-performance supercapacitor electrodes prepared from dispersions of tetrabenzonaphthalene-based conjugated microporous polymers and carbon nanotubes. ACS Applied Materials and Interfaces, 13, 51906–51916 (2021)

    Article  Google Scholar 

  9. LEE, W. S. and CHOI, J. Hybrid integration of carbon nanotubes and transition metal dichalcogenides on cellulose paper for highly sensitive and extremely deformable chemical sensors. ACS Applied Materials and Interfaces, 11, 19363–19371 (2019)

    Article  Google Scholar 

  10. ZHANG, J. and WANG, C. Buckling of carbon honeycombs: a new mechanism for molecular mass transportation. The Journal of Physical Chemistry C, 121, 8196–8203 (2017)

    Article  Google Scholar 

  11. GENOESE, A., GENOESE, A., and SALERNO, G. Buckling and post-buckling analysis of single wall carbon nanotubes using molecular mechanics. Applied Mathematical Modelling, 83, 777–800 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. WANG, C. M., ZHANG, Y. Y., XIANG, Y., and REDDY, J. N. Recent studies on buckling of carbon nanotubes. Applied Mechanics Reviews, 63, 030804 (2010)

    Article  Google Scholar 

  13. SILVESTRE, N., FARIA, B., and CANONGIA LOPES, J. N. A molecular dynamics study on the thickness and post-critical strength of carbon nanotubes. Composite Structures, 94, 1352–1358 (2012)

    Article  Google Scholar 

  14. XU, X. J. and DENG, Z. C. Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory. Applied Mathematics and Mechanics (English Edition), 35(9), 1115–1128 (2014) https://doi.org/10.1007/s10483-014-1855-6

    Article  MathSciNet  MATH  Google Scholar 

  15. WANG, J. F., SHI, S. Q., YANG, J. P., and ZHANG, W. Multiscale analysis on free vibration of functionally graded graphene reinforced PMMA composite plates. Applied Mathematical Modelling, 98, 38–58 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  16. YAKOBSON, B. I., BRABEC, C. J., and BERNHOLC, J. Nanomechanics of carbon tubes: instabilities beyond linear response. Physical Review Letters, 76, 2511–2514 (1996)

    Article  Google Scholar 

  17. RU, C. Q. Effective bending stiffness of carbon nanotubes. Physical Review B, 62, 9973–9976 (2000)

    Article  Google Scholar 

  18. BIAN, L. C. and WANG, Y. W. Temperature-related study on buckling properties of double-walled carbon nanotubes. European Journal of Mechanics-A/Solids, 80, 103875 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. MOHAMED, N., MOHAMED, S. A., and ELTAHER, M. A. Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model. Engineering with Computers, 37, 2823–2836 (2020)

    Article  Google Scholar 

  20. HE, X. Q., KITIPORNCHAI, S., and LIEW, K. M. Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction. Journal of the Mechanics and Physics of Solids, 53, 303–326 (2005)

    Article  MATH  Google Scholar 

  21. SILVESTRE, N., WANG, C. M., ZHANG, Y. Y., and XIANG, Y. Sanders shell model for buckling of single-walled carbon nanotubes with small aspect ratio. Composite Structures, 93, 1683–1691 (2011)

    Article  Google Scholar 

  22. LEISSA, A. W. Vibration of Shells, NASA, Washington, D. C. (1973)

    MATH  Google Scholar 

  23. LOUHGHALAM, A., IGUSA, T., and TOOTKABONI, M. Dynamic characteristics of laminated thin cylindrical shells: asymptotic analysis accounting for edge effect. Composite Structures, 112, 22–37 (2014)

    Article  Google Scholar 

  24. JAUNKY, N. and KNIGHT, N. F. An assessment of shell theories for buckling of circular cylindrical laminated composite panels loaded in axial compression. International Journal of Solids and Structures, 36, 3799–3820 (1999)

    Article  MATH  Google Scholar 

  25. GULYAEV, V. I., LUGOVOI, P. Z., and LYSYUK, N. A. Propagation of harmonic waves in a cylindrical shell (Timoshenko model). International Applied Mechanics, 39, 472–478 (2003)

    Article  MATH  Google Scholar 

  26. XIANG, Y., WANG, C. M., LIM, C. W., and KITIPORNCHAI, S. Buckling of intermediate ring supported cylindrical shells under axial compression. Thin-Walled Structures, 43, 427–443 (2005)

    Article  Google Scholar 

  27. STROZZI, M., ELISHAKOFF, I. E., MANEVITCH, L. I., and GENDELMAN, O. V. Applicability and limitations of Donnell shell theory for vibration modelling of double-walled carbon nanotubes. Thin-Walled Structures, 178, 109532 (2022)

    Article  Google Scholar 

  28. TIMESLI, A., BRAIKAT, B., JAMAL, M., and DAMIL, N. Prediction of the critical buckling load of multi-walled carbon nanotubes under axial compression. Comptes Rendus Mécanique, 345, 158–168 (2017)

    Article  Google Scholar 

  29. GUPTA, S., PRAMANIK, S., SMITA, DAS, S. K., and SAHA, S. Dynamic analysis of wave propagation and buckling phenomena in carbon nanotubes (CNTs). Wave Motion, 104, 102730 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  30. HE, X. Q., QU, C., QIN, Q. H., and WANG, C. M. Buckling and postbuckling analysis of multi-walled carbon nanotubes based on the continuum shell model. International Journal of Structural Stability and Dynamics, 7, 629–645 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. YAO, X. and HAN, Q. Postbuckling prediction of double-walled carbon nanotubes under axial compression. European Journal of Mechanics-A/Solids, 26, 20–32 (2007)

    Article  MATH  Google Scholar 

  32. SUN, Y., YAO, X., and HAN, Q. Combined torsional buckling of double-walled carbon nanotubes with axial load in the multi-field coupled condition. Science China-Physics Mechanics & Astronomy, 54, 1659–1665 (2011)

    Article  Google Scholar 

  33. SUN, C., LIU, K., and HONG, Y. Dynamic shell buckling behavior of multi-walled carbon nanotubes embedded in an elastic medium. Science China-Physics Mechanics & Astronomy, 56, 483–490 (2013)

    Article  Google Scholar 

  34. GARG, A., CHALAK, H. D., BELARBI, M. O., ZENKOUR, A. M., and SAHOO, R. Estimation of carbon nanotubes and their applications as reinforcing composite materials: an engineering review. Composite Structures, 272, 114234 (2021)

    Article  Google Scholar 

  35. RU, C. Q. Axially compressed buckling of a doublewalled carbon nanotube embedded in an elastic medium. Journal of the Mechanics and Physics of Solids, 49, 1265–1279 (2001)

    Article  MATH  Google Scholar 

  36. WANG, C. Y., RU, C. Q., and MIODUCHOWSKI, A. Axially compressed buckling of pressured multiwall carbon nanotubes. International Journal of Solids and Structures, 40, 3893–3911 (2003)

    Article  MATH  Google Scholar 

  37. WANG, J. F. and ZHANG, W. An equivalent continuum meshless approach for material nonlinear analysis of CNT-reinforced composites. Composite Structures, 188, 116–125 (2018)

    Article  Google Scholar 

  38. GHORBANPOUR ARANI, A., RAHMANI, R., AREFMANESH, A., and GOLABI, S. Buckling analysis of multi-walled carbon nanotubes under combined loading considering the effect of small length scale. Journal of Mechanical Science and Technology, 22, 429–439 (2008)

    Article  Google Scholar 

  39. HE, X. Q., KITIPORNCHAI, S., WANG, C. M., XIANG, Y., and ZHOU, Q. A nonlinear van der Waals force model for multiwalled carbon nanotubes modeled by a nested system of cylindrical shells. Journal of Applied Mechanics, 77, 061006 (2010)

    Article  Google Scholar 

  40. SAITO, R., DRESSELHAUS, G., and DRESSELHAUS, M. S. Physical Properties of Carbon Nanotubes, Imperial College Press, London (1998)

    Book  MATH  Google Scholar 

  41. HARIK, V. M. Mechanics of carbon nanotubes: applicability of the continuum-beam models. Computational Materials Science, 24, 328–342 (2002)

    Article  Google Scholar 

  42. SUN, C. Q., LIU, K. X., and HONG, Y. S. Axisymmetric compressive buckling of multi-walled carbon nanotubes under different boundary conditions. Acta Mechanica Sinica, 28, 83–90 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. RU, C. Q. Effect of van der Waals forces on axial buckling of a double-walled carbon nanotube. Journal of Applied Physics, 87, 7227–7231 (2000)

    Article  Google Scholar 

  44. SAITO, R., MATSUO, R., KIMURA, T., DRESSELHAUS, G., and DRESSELHAUS, M. S. Anomalous potential barrier of double-wall carbon nanotube. Chemical Physics Letters, 348, 187–193 (2001)

    Article  Google Scholar 

  45. HE, X. Q., KITIPORNCHAI, S., WANG, C. M., and LIEW, K. M. Modeling of van der Waals force for infinitesimal deformation of multi-walled carbon nanotubes treated as cylindrical shells. International Journal of Solids and Structures, 42, 6032–6047 (2005)

    Article  MATH  Google Scholar 

  46. AMABILI, M. A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach. Journal of Sound and Vibration, 264, 1091–1125 (2003)

    Article  Google Scholar 

  47. LIEW, K. M., WONG, C. H., HE, X. Q., TAN, M. J., and MEGUID, S. A. Nanomechanics of single and multiwalled carbon nanotubes. Physical Review B, 69, 115429 (2004)

    Article  Google Scholar 

  48. SHI, J. X., NATSUKI, T., and NI, Q. Q. Radial buckling of multi-walled carbon nanotubes under hydrostatic pressure. Applied Physics A-Materials Science & Processing, 117, 1103–1108 (2014)

    Article  Google Scholar 

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Authors and Affiliations

  1. Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, Faculty of Materials and Manufacturing, Beijing University of Technology, Beijing, 100124, China

    Xinlei Li & Jianfei Wang

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  1. Xinlei Li
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  2. Jianfei Wang
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Correspondence to Jianfei Wang.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

Conflict of Interests

The authors declare that there is no conflict of interest regarding the publication of this paper.

Citation: LI, X. L. and WANG, J. F. Effects of layer number and initial pressure on continuum-based buckling analysis of multi-walled carbon nanotubes accounting for van der Waals interaction. Applied Mathematics and Mechanics (English Edition), 43(12), 1857–1872 (2022) https://doi.org/10.1007/s10483-022-2909-6

Project supported by the National Natural Science Foundation of China (No. 12072003) and the Beijing Natural Science Foundation of China (No. 1222001)

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Li, X., Wang, J. Effects of layer number and initial pressure on continuum-based buckling analysis of multi-walled carbon nanotubes accounting for van der Waals interaction. Appl. Math. Mech.-Engl. Ed. 43, 1857–1872 (2022). https://doi.org/10.1007/s10483-022-2909-6

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  • DOI: https://doi.org/10.1007/s10483-022-2909-6

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