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August 4, 2023
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August 4, 2023
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Abstract

A single-elastic beam model is being developed based on Eringens nonlocal elasticity and Timoshenko beam theory. Nonlocal parameter takes into account the small scale effects in the analysis of nano-size structures. Derived herein is a decoupled sixth-order nonlocal governing differential equations for the frequency and stability analysis of nano-scale short beams. The effect of shear deformation is thereby included in the small scale analysis. A Differential Quadrature (DQ) approach is being applied and its elaborate method of solution is illustrated. The higher order DQ approach is found to be a good numerical technique for the convenient and rapid solution of the aforementioned nonlocal problems. Frequency and buckling results for various scale-based nonlocal parameters are shown. Effect of number of interpolation points on the accuracy of the results is also investigated. It is seen that there is a significant effect of aspect ratio and nonlocal parameter on the nonlocal frequency and buckling loads of nano-scale beams. Also, the present study could open up a new approach of solution technique for the analysis of nano-scale structures based on nonlocal Timoshenko theory.

Keywords

Nonlocal elasticity, classical mechanics, Timoshenko beam, vibration, buckling and differential quadrature.

Article Details

How to Cite
Murmu, T., & Pradhan, S. (2023). Vibration and Buckling Analysis of Nano-Scale Beams via Nonlocal Elasticity and Timoshenko Beam Theory : A Differential Quadrature Approach. Journal of Aerospace Sciences and Technologies, 62(1), 40–54. https://doi.org/10.61653/joast.v62i1.2010.486
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References

  1. Lu. P., "Dynamic Analysis of Axially Prestressed Micro/nanobeam Structures Based on Nonlocal Beam Theory", Journal of Applied Physics, 2007, Vol.101, 073504.
  2. Wang, C. M. and Duan, W. H., "Free Vibration of Nanorings/arches Based on Nonlocal Elasticity", Journal of Applied Physics, 2008, Vol. 104, 014303.
  3. Yuan, Z. Y., Colomer, J. F. and Su, B. L., "Titanium Oxide Nanoribbons", Chemical Physics Letters, 2002, Vol. 363 (3-4), pp. 362-366.
  4. Duan, W. H. and Wang, C. M., "Exact Solutions for Axisymmetric Bending of Micro/nanoscale Circular Plates Based on Nonlocal Plate Theory", Nanotechnology, 2007, Vol. 18, 385704 (5pp).
  5. Yoon, J, Ru, C. Q. and Mioduchowski, A., "Vibration and Instability of Carbon Nanotubes Conveying Fluid", Composites Science and Technology, 2005, Vol. 65, pp.1326-1336.
  6. Wang, L., Ni, Q., Li, M. and Qian, Q., "The Thermal Effect on Vibration and Instability of Carbon Nanotubes Conveying Fluid", Physica E, 2008, Vol. 40, 2008, pp .3179-3182.
  7. Globus, A.I., "Molecular Nanotechnology in Aerospace: 1999", NASA Technical Report, 2000, NAS- 00-001.
  8. Eringen, A C., "Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves", International Journal of Engineering Science, 1972, Vol.10, pp.1-16.
  9. Eringen, A. C., "On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves", Journal of Applied physics, 1983, Vol.54, pp. 4703- 4710.
  10. Park, J. M., Kim, P. G., Jang, J. H., Wang. Z., Kim. J. W., Lee, W., Park, J. G. and DeVries, K. L., "Self-sensing and Dispersive Evaluation of Single Carbon Fiber/Carbon Nanotubes (CNT) -Epoxy Composites Using Electro-micromechanical Technique and Nondestructive Acoustic Emission", 2008, Composites Part B: Engineering, Vol. 39 (7-8), pp. 1170-1182.
  11. Peddieson, J., Buchanan. G. G. and McNitt, R. P., "Application of Nonlocal Continuum Models to Nanotechnology", International Journal of Engineering Science, 2003, Vol.41, pp. 305-312. 12. Sudak, L. J., "Column Buckling of Multiwalled Carbon Nanotubes Using Nonlocal Continuum Mechanics", Journal of Applied Physics, 2003, Vol. 94, pp.7281-7287.
  12. Wang, Q., Varadan, V. K. and Quek, S. T., "Small Scale Effect on Elastic Buckling of Carbon Nanotubes with Nonlocal Continuum Models", Physics letters A, 2006, Vol. 357, pp. 130-135.
  13. Lu, P., Lee, H. P., Lu, C. and Zhang, P. Q., "Dynamic Properties of Flexural Beams Using a Nonlocal Elasticity Model", Journal of Applied Physics, 2006, Vol.99, 073510.
  14. Reddy, J. N., "Nonlocal Theories for Bending, Buckling and Vibration of Beams", International Journal of Engineering Science, 2007, Vol. 45, pp. 288-307.
  15. Pradhan, S. C. and Murmu, T., "Differential Quadrature Method for Vibration Analysis of Beam on Winkler Foundation Based on Nonlocal Elastic Theory", Journal of Institution of Engineers (India), Mechanical Division, 2009, Vol 89, pp.3-12.
  16. Murmu, T. and Pradhan, S. C., "Buckling Analysis of Beam on Winkler Foundation by using MDQM and Nonlocal Theory", Journal of Aerospace Sciences and Technologies, 2008, Vol. 60 (3), pp. 206- 215.
  17. Bellman, R. E. and Casti, J., "Differential Quadrature and Long-term Integration", Journal of Mathematical Analysis and Applications, 1971, Vol.34, pp. 235-238.
  18. Bellman, R., Kashef, B.G. and Casti, J., "Differential Quadrature: A Technique for a Rapid Solution of Non Linear Partial Differential Equations", Journal of Computational Physics, 1972, Vol.10, pp. 40-52.
  19. Civan, F. and Sliepcevich, C. M., "On the Solution of the Thomas-Fermi Equation by Differential Quadrature", Journal of Computational Physics, 1984, Vol. 56, pp. 343-348.
  20. Bert, C. W., Jang, S. K. and Striz, A. G., "Two New Approximate Methods for Analyzing Free Vibration of Structural Components", AIAA Journal. 1988, Vol. 26, pp. 612-618.
  21. Jang, S. K., Bert, C. W. and. Striz, A. G. , "Application of Differential Quadrature to Static Analysis of Structural Components", International Journal of Numerical Methods in Engineering, 1989, Vol.28, pp. 561-577.
  22. Pradhan, S. C. and Murmu, T., "Thermo-mechanical Vibration of FGM Sandwich Beam Under Variable Elastic Foundation Using Differential Quadrature Method", Journal of Sound and Vibration, 2009, Vol.321 (1-2), pp. 342 -362.
  23. Mohammad, R. H., Mohammad, J. A. and Malekzadeh, P., "A Differential Quadrature Analysis of Unsteady Open Channel Flow", Applied Mathematical Modeling, 2007, Vol. 31(8), pp. 1594 -1608.
  24. Pradhan, S. C. and Murmu, T., "Differential Quadrature Approach for Dynamic Analysis of Beams Based on Nonlocal Elasticity Model", Proceedings of Fifth International Conference on Smart Materials, Structures and Systems (ISSS), 24-26 July, 2008, Indian Institute of Science, Bangalore.
  25. Chen, Y., Lee, J. D. and Eskandarian, A., "Atomistic Viewpoint of the Applicability of Microcontinuum Theories", International Journal of Solids and Structures, 2004, Vol. 41, pp. 2085-2097.
  26. Wang, L.F. and Hu, H.Y., "Flexural Wave Propagation in Single-walled Carbonnanotubes", Physical Review B, 2005, Vol. 71, 195412.
  27. Wang, Q., Han, Q. K. and Wen, B. C., "Estimate of Material Property of Carbon Nanotubes via Nonlocal Elasticity", Advanced Theoretical Applied Mechanics, 2008, Vol.1 pp.1-10.
  28. Sondergaard, N., Ghatnekar-Nilsson, S., Guhr, T. and Montelius, L., "Elasticity of Macroscopic Isotropic Media Applied to Nano-cantilevers", Cond- Mat. Mtrl-sci/0603696v, 2006, pp.1-10.
  29. Iijima, S., "Heliacl Microtubules of Graphitic Carbon", Nature 1991, Vol. pp. 56-58.
  30. Wang, Q. and Wang, C. M., "The Constitutive Relation and Small Scale Parameter of Nonlocal Continuum Mechanics for Modeling Carbon Nanotubes, Nanotechnology, 2007, Vol.18, 075702 .