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Static, Buckling, and Free Vibration Analysis of CNT Reinforced Composite Beams with Elastic Foundation Using IHSDT

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Abstract

Purpose

A carbon nanotube-reinforced composite (CNTRC) beam resting on a Pasternak elastic foundation which consists of a Winkler spring and shear layer is investigated to obtain the bending, buckling, and free vibration responses using inverse hyperbolic shear deformation theory (IHSDT). The shear strain shape function is employed in this study to construct a nonlinear distribution of transverse shear stresses. The theory fulfills the traction-free boundary conditions on both the upper and lower surfaces of the beam, hence no shear correction factor is needed.

Methods

Hamilton’s principle is employed to derive the equation of motion and Navier’s solution technique is used to determining the closed-from solution for the CNTRC beam on the Pasternak foundation. To determine the material properties of CNTRC beams, the rule of mixture is used. In this study, various types of CNT reinforcement distribution are used such as uniform distribution (UD-Beam), X-Beam, O-Beam, and V-Beam.

Results

The deformation, stresses, critical buckling load, and natural frequencies of the simply supported CNTRC beam resting on the Pasternak elastic foundation are investigated using an analytical approach, that takes into account various length-to-thickness ratios, CNT volume fraction, CNT distribution, Winkler spring constant factor, and shear layer constant factor.

Conclusion

The present theory predicts the structural responses quite accurately compared to the available theories in the literature. Some new results are also included for the benchmark solutions for the new research.

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References

  1. Iijima S (1991) Helical microtubules of graphitic carbon. Nature 354(6348):56–58. https://doi.org/10.1038/354056a0

    Article  Google Scholar 

  2. Khaniki HB, Ghayesh MH (2020) A review on the mechanics of carbon nanotube strengthened deformable structures. Eng Struct. https://doi.org/10.1016/J.ENGSTRUCT.2020.110711

    Article  Google Scholar 

  3. Lau AKT, Hui D (2002) The revolutionary creation of new advanced materials—carbon nanotube composites. Compos B Eng 33:263–277. https://doi.org/10.1016/S1359-8368(02)00012-4

    Article  Google Scholar 

  4. Thostenson ET, Ren Z, Chou TW (2001) Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol 61:1899–1912. https://doi.org/10.1016/S0266-3538(01)00094-X

    Article  Google Scholar 

  5. Saito Y, Hamaguchi K, Hata K et al (1997) Conical beams from open nanotubes. Nature 389(6651):554–555. https://doi.org/10.1038/39221

    Article  Google Scholar 

  6. Niu C, Sichel EK, Hoch R et al (1997) High power electrochemical capacitors based on carbon nanotube electrodes. Appl Phys Lett 70:1480–1482. https://doi.org/10.1063/1.118568

    Article  Google Scholar 

  7. Vaccarini L, Goze C, Henrard L et al (2000) Mechanical and electronic properties of carbon and boron–nitride nanotubes. Carbon N Y 38:1681–1690. https://doi.org/10.1016/S0008-6223(99)00293-6

    Article  Google Scholar 

  8. Lijima S, Aikawa Y, Baba K (1991) Growth of diamond particles in chemical vapor deposition. J Mater Res 6(7):1491–1497. https://doi.org/10.1557/JMR.1991.1491

    Article  Google Scholar 

  9. Iijima S, Ichihashi T (1993) Single-shell carbon nanotubes of 1-nm diameter. Nature 363(6430):603–605. https://doi.org/10.1038/363603a0

    Article  Google Scholar 

  10. Bethune DS, Klang CH, De Vries MS et al (1993) Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls. Nature 363(6430):605–607. https://doi.org/10.1038/363605a0

    Article  Google Scholar 

  11. Kelly BT (1981) Physics of graphite 477

  12. Liew KM, Pan Z, Zhang L-W et al (2020) The recent progress of functionally graded CNT reinforced composites and structures. SCPMA 63:234601. https://doi.org/10.1007/S11433-019-1457-2

    Article  Google Scholar 

  13. Liew KM, Pan ZZ, Zhang LW (2019) An overview of layerwise theories for composite laminates and structures: development, numerical implementation and application. Compos Struct 216:240–259. https://doi.org/10.1016/J.COMPSTRUCT.2019.02.074

    Article  Google Scholar 

  14. Imani Yengejeh S, Kazemi SA, Öchsner A (2017) Carbon nanotubes as reinforcement in composites: a review of the analytical, numerical and experimental approaches. Comput Mater Sci 136:85–101. https://doi.org/10.1016/J.COMMATSCI.2017.04.023

    Article  Google Scholar 

  15. Ajayan PM, Stephan O, Colliex C, Trauth D (1994) Aligned carbon nanotube arrays formed by cutting a polymer resin–nanotube composite. Science 265:1212–1214. https://doi.org/10.1126/SCIENCE.265.5176.1212

    Article  Google Scholar 

  16. Odegard GM, Gates TS, Wise KE et al (2003) Constitutive modeling of nanotube–reinforced polymer composites. Compos Sci Technol 63:1671–1687. https://doi.org/10.1016/S0266-3538(03)00063-0

    Article  Google Scholar 

  17. Griebel M, Hamaekers J (2004) Molecular dynamics simulations of the elastic moduli of polymer–carbon nanotube composites. Comput Methods Appl Mech Eng 193:1773–1788. https://doi.org/10.1016/J.CMA.2003.12.025

    Article  MathSciNet  Google Scholar 

  18. Mokashi VV, Qian D, Liu Y (2007) A study on the tensile response and fracture in carbon nanotube-based composites using molecular mechanics. Compos Sci Technol 67:530–540. https://doi.org/10.1016/J.COMPSCITECH.2006.08.014

    Article  Google Scholar 

  19. Lv T, Yao Y, Li N, Chen T (2016) Highly stretchable supercapacitors based on aligned carbon nanotube/molybdenum disulfide composites. Angew Chem Int Ed 55:9191–9195. https://doi.org/10.1002/ANIE.201603356

    Article  Google Scholar 

  20. Yasuda S, Furuya A, Uchibori Y et al (2016) Iron–nitrogen-doped vertically aligned carbon nanotube electrocatalyst for the oxygen reduction reaction. Adv Funct Mater 26:738–744. https://doi.org/10.1002/ADFM.201503613

    Article  Google Scholar 

  21. Xie H, Zhang R, Zhang Y et al (2016) Preloading catalysts in the reactor for repeated growth of horizontally aligned carbon nanotube arrays. Carbon N Y 98:157–161. https://doi.org/10.1016/J.CARBON.2015.11.001

    Article  Google Scholar 

  22. Kang L, Zhang S, Li Q, Zhang J (2016) Growth of horizontal semiconducting SWNT arrays with density higher than 100 tubes/μm using ethanol/methane chemical vapor deposition. J Am Chem Soc 138:6727–6730. https://doi.org/10.1021/JACS.6B03527/ASSET/IMAGES/LARGE/JA-2016-03527H_0004.JPEG

    Article  Google Scholar 

  23. Goh GL, Agarwala S, Yeong WY (2019) Directed and on-demand alignment of carbon nanotube: a review toward 3D printing of electronics. Adv Mater Interfaces 6:1801318. https://doi.org/10.1002/ADMI.201801318

    Article  Google Scholar 

  24. Iakoubovskii K (2009) Techniques of aligning carbon nanotubes. Cent Eur J Phys 7:645–653. https://doi.org/10.2478/S11534-009-0072-2/MACHINEREADABLECITATION/RIS

    Article  Google Scholar 

  25. Beigmoradi R, Samimi A, Mohebbi-Kalhori D (2018) Engineering of oriented carbon nanotubes in composite materials. Beilstein J Nanotechnol 9(41):415–435. https://doi.org/10.3762/BJNANO.9.41

    Article  Google Scholar 

  26. Fisher FT, Bradshaw RD, Brinson LC (2002) Effects of nanotube waviness on the modulus of nanotube-reinforced polymers. Appl Phys Lett 80:4647. https://doi.org/10.1063/1.1487900

    Article  Google Scholar 

  27. Xie S, Li W, Pan Z et al (2000) Mechanical and physical properties on carbon nanotube. J Phys Chem Solids 61:1153–1158. https://doi.org/10.1016/S0022-3697(99)00376-5

    Article  Google Scholar 

  28. Ruoff RS, Lorents DC (1995) Mechanical and thermal properties of carbon nanotubes. Carbon N Y 33:925–930. https://doi.org/10.1016/0008-6223(95)00021-5

    Article  Google Scholar 

  29. Britto PJ, Santhanam KSV, Rubio A et al (1999) Improved charge transfer at carbon nanotube electrodes. Adv Mater. https://doi.org/10.1002/(SICI)1521-4095(199902)11:2

    Article  Google Scholar 

  30. Chen P, Wu X, Lin J (1979) Tan KL (1999) High H2 uptake by alkali-doped carbon nanotubes under ambient pressure and moderate temperatures. Science 285:91–93. https://doi.org/10.1126/SCIENCE.285.5424.91/ASSET/F5A52D4F-0F6B-42FC-AD53-1B1E13889218/ASSETS/GRAPHIC/SE2597628002.JPEG

    Article  Google Scholar 

  31. Liu C, Fan YY, Liu M et al (1999) Hydrogen storage in single-walled carbon nanotubes at room temperature. Science 286:1127–1129. https://doi.org/10.1126/SCIENCE.286.5442.1127/ASSET/3583E91A-2068-4C36-AF1E-E45671795BB4/ASSETS/GRAPHIC/SE4397962003.JPEG

    Article  Google Scholar 

  32. Samuel Ratna Kumar PS, Robinson Smart DS, John Alexis S (2017) Corrosion behaviour of aluminium metal matrix reinforced with multi-wall carbon nanotube. J Asian Ceram Soc 5:71–75. https://doi.org/10.1016/J.JASCER.2017.01.004

    Article  Google Scholar 

  33. Shah KA, Tali BA (2016) Synthesis of carbon nanotubes by catalytic chemical vapour deposition: a review on carbon sources, catalysts and substrates. Mater Sci Semicond Process 41:67–82. https://doi.org/10.1016/J.MSSP.2015.08.013

    Article  Google Scholar 

  34. Arora N, Sharma NN (2014) Arc discharge synthesis of carbon nanotubes: comprehensive review. Diam Relat Mater 50:135–150. https://doi.org/10.1016/J.DIAMOND.2014.10.001

    Article  Google Scholar 

  35. Su Y, Zhang Y (2015) Carbon nanomaterials synthesized by arc discharge hot plasma. Carbon 83:90–99. https://doi.org/10.1016/J.CARBON.2014.11.023

    Article  Google Scholar 

  36. Harris PJF (2007) Solid state growth mechanisms for carbon nanotubes. Carbon 45:229–239. https://doi.org/10.1016/J.CARBON.2006.09.023

    Article  Google Scholar 

  37. Kingston CT, Simard B (2006) Recent advances in laser synthesis of single-walled carbon nanotubes. J Nanosci Nanotechnol 6:1225–1232. https://doi.org/10.1166/JNN.2006.310

    Article  Google Scholar 

  38. Arepalli S (2004) Laser ablation process for single-walled carbon nanotube production. J Nanosci Nanotechnol 4:317–325. https://doi.org/10.1166/JNN.2004.072

    Article  Google Scholar 

  39. Mubarak NM, Abdullah EC, Jayakumar NS, Sahu JN (2014) An overview on methods for the production of carbon nanotubes. J Ind Eng Chem 20:1186–1197. https://doi.org/10.1016/J.JIEC.2013.09.001

    Article  Google Scholar 

  40. Jourdain V, Bichara C (2013) Current understanding of the growth of carbon nanotubes in catalytic chemical vapour deposition. Carbon N Y. https://doi.org/10.1016/j.carbon.2013.02.046ï

    Article  Google Scholar 

  41. Koziol K, Boskovic BO, Yahya N (2011) Synthesis of carbon nanostructures by CVD method. Adv Struct Mater 5:23–49. https://doi.org/10.1007/8611_2010_12/COVER

    Article  Google Scholar 

  42. Han Y, Elliott J (2007) Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput Mater Sci 39:315–323. https://doi.org/10.1016/J.COMMATSCI.2006.06.011

    Article  Google Scholar 

  43. Wattanasakulpong N, Ungbhakorn V (2013) Analytical solutions for bending, buckling and vibration responses of carbon nanotube-reinforced composite beams resting on elastic foundation. Comput Mater Sci 71:201–208. https://doi.org/10.1016/J.COMMATSCI.2013.01.028

    Article  Google Scholar 

  44. Wuite J, Adali S (2005) Deflection and stress behaviour of nanocomposite reinforced beams using a multiscale analysis. Compos Struct 71:388–396. https://doi.org/10.1016/j.compstruct.2005.09.011

    Article  Google Scholar 

  45. Shen HS, Xiang Y (2013) Nonlinear analysis of nanotube-reinforced composite beams resting on elastic foundations in thermal environments. Eng Struct 56:698–708. https://doi.org/10.1016/J.ENGSTRUCT.2013.06.002

    Article  Google Scholar 

  46. Kumar P, Srinivas J (2017) Free vibration, bending and buckling of a FG-CNT reinforced composite beam: comparative analysis with hybrid laminated composite beam. Multidiscip Model Mater Struct 13:590–611. https://doi.org/10.1108/MMMS-05-2017-0032/FULL/PDF

    Article  Google Scholar 

  47. Yas MH, Samadi N (2012) Free vibrations and buckling analysis of carbon nanotube-reinforced composite Timoshenko beams on elastic foundation. Int J Press Vessels Pip 98:119–128. https://doi.org/10.1016/J.IJPVP.2012.07.012

    Article  Google Scholar 

  48. Wu HL, Yang J, Kitipornchai S (2016) Imperfection sensitivity of postbuckling behaviour of functionally graded carbon nanotube-reinforced composite beams. Thin-Walled Struct 108:225–233. https://doi.org/10.1016/J.TWS.2016.08.024

    Article  Google Scholar 

  49. Kiani Y (2016) Thermal postbuckling of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets. J Therm Stresses 39:1098–1110. https://doi.org/10.1080/01495739.2016.1192856

    Article  Google Scholar 

  50. Mirzaei M, Kiani Y (2016) Nonlinear free vibration of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets. Acta Mech 227(7):1869–1884. https://doi.org/10.1007/S00707-016-1593-6

    Article  MathSciNet  Google Scholar 

  51. Singh SD, Sahoo R (2021) Static and free vibration analysis of functionally graded CNT reinforced sandwich plates using inverse hyperbolic shear deformation theory. J Strain Anal Eng Des 56:386–403. https://doi.org/10.1177/0309324720957568/ASSET/IMAGES/LARGE/10.1177_0309324720957568-FIG2.JPEG

    Article  Google Scholar 

  52. Singh SD, Sahoo R (2020) Static and free vibration analysis of functionally graded CNT reinforced composite plates using trigonometric shear deformation theory. Structures 28:685–696. https://doi.org/10.1016/J.ISTRUC.2020.09.008

    Article  Google Scholar 

  53. Singh SD, Sahoo R (2021) Analytical solution for static and free vibration analysis of functionally graded CNT-reinforced sandwich plates. Arch Appl Mech 91:3819–3834. https://doi.org/10.1007/S00419-021-01979-1/FIGURES/6

    Article  Google Scholar 

  54. Xu H, Wang YQ (2022) Differential transformation method for free vibration analysis of rotating Timoshenko beams with elastic boundary conditions. Int J Appl Mech. https://doi.org/10.1142/S1758825122500466

    Article  Google Scholar 

  55. Cui Y, Wang Y (2024) Effect of disk flexibility on nonlinear vibration characteristics of shaft-disk rotors. Acta Mech Sinica/Lixue Xuebao 40:1–15. https://doi.org/10.1007/S10409-023-23140-X/METRICS

    Article  MathSciNet  Google Scholar 

  56. Wang YQ, Xing WC, Wang J, Chai Q (2023) Theoretical and experimental studies on vibration characteristics of bolted joint multi-plate structures. Int J Mech Sci 252:108348. https://doi.org/10.1016/J.IJMECSCI.2023.108348

    Article  Google Scholar 

  57. Xing WC, Wang J, Wang YQ (2024) An effective model for bolted flange joints and its application in vibrations of bolted flange joint multiple-plate structures: theory with experiment verification. Appl Math Model 126:482–505. https://doi.org/10.1016/J.APM.2023.11.013

    Article  MathSciNet  Google Scholar 

  58. Grover N, Maiti DK, Singh BN (2013) A new inverse hyperbolic shear deformation theory for static and buckling analysis of laminated composite and sandwich plates. Compos Struct 95:667–675. https://doi.org/10.1016/J.COMPSTRUCT.2012.08.012

    Article  Google Scholar 

  59. Ke LL, Yang J, Kitipornchai S (2010) Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams. Compos Struct 92:676–683. https://doi.org/10.1016/J.COMPSTRUCT.2009.09.024

    Article  Google Scholar 

  60. Reddy JN (1984) A simple higher-order theory for laminated composite plates. J Appl Mech 51:745–752. https://doi.org/10.1115/1.3167719

    Article  Google Scholar 

  61. Tagrara SH, Benachour A, Bouiadjra MB et al (2015) On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams. Steel Compos Struct 19:1259. https://doi.org/10.12989/SCS.2015.19.5.1259

    Article  Google Scholar 

  62. Liew KM, Yang J, Kitipornchai S (2003) Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading. Int J Solids Struct 40:3869–3892. https://doi.org/10.1016/S0020-7683(03)00096-9

    Article  Google Scholar 

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Acknowledgements

The support of Science and Engineering Research Board (SERB), Department of Science and Technology (DST), India under grant number SPG/2021/001682 is acknowledged.

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  1. Department of Civil Engineering, Indian Institute of Technology (BHU), Varanasi, 221005, India

    Abhijeet Babasaheb Babar & Rosalin Sahoo

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  1. Abhijeet Babasaheb Babar
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ABB: Visualization, Methodology, Validation, Writing – original draft. RS: Conceptualization, Supervision, Writing – review & editing.

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Correspondence to Rosalin Sahoo.

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Babar, A.B., Sahoo, R. Static, Buckling, and Free Vibration Analysis of CNT Reinforced Composite Beams with Elastic Foundation Using IHSDT. J. Vib. Eng. Technol. (2024). https://doi.org/10.1007/s42417-024-01349-5

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