A COUPLED MCST-FEM INVESTIGATION OF SIZE-DEPENDENT BUCKLING OF PERFORATED NANOBEAMS ON WINKLER-PASTERNAK FOUNDATION

Abstract

The buckling behavior of perforated nanobeams on elastic foundations has become increasingly important, mainly due to their widespread use in nanostructures and nanotechnology systems. This study investigates the buckling behavior of perforated nanobeams resting on Winkler-Pasternak elastic foundations using Modified Couple Stress Theory (MCST) and the Finite Element Method (FEM). The analysis examines the effects of various parameters, including foundation elasticity, MCST internal length scale, perforation properties, and beam length, on critical buckling loads. Results indicate that increasing both Winkler and Pasternak foundation parameters enhances the critical buckling load, with the Pasternak parameter showing a more pronounced effect due to its incorporation of shear effects. The MCST internal length scale parameter significantly influences nano-beam stability, highlighting the importance of size effects at nanoscale dimensions. Higher filling ratios correlate directly with increased buckling resistance, while a greater number of holes reduces overall structural stiffness and decreases the critical buckling load. Beam length exhibits an inverse relationship with buckling strength; longer beams demonstrate lower critical buckling loads than shorter beams, regardless of the number of holes present.

Keywords

• Buckling Analysis
• Finite Element Method
• Modified Couple Stress Theory
• Perforated Nanobeam
• Winkler-Pasternak Foundation

References

1. M. A. Eltaher, A. Khairy, A. M. Sadoun, and F.-A. Omar, “Static and buckling analysis of functionally graded Timoshenko nanobeams,” Appl Math Comput, vol. 229, pp. 283–295, Feb. 2014, doi: 10.1016/j.amc.2013.12.072.
2. A. A. Abdelrahman and M. A. Eltaher, “On bending and buckling responses of perforated nanobeams including surface energy for different beams theories,” Eng Comput, vol. 38, no. 3, pp. 2385–2411, Jun. 2022, doi: 10.1007/s00366-020-01211-8.
3. M. Ö. Yaylı, “Stability analysis of a rotationally restrained microbar embedded in an elastic matrix using strain gradient elasticity,” Curved and Layered Structures, vol. 6, no. 1, pp. 1–10, Jan. 2019, doi: 10.1515/cls-2019-0001.
4. F. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong, “Couple stress based strain gradient theory for elasticity,” Int J Solids Struct, vol. 39, no. 10, pp. 2731–2743, May 2002, doi: 10.1016/S0020-7683(02)00152-X.
5. M. Ö. Yaylı, “Torsional vibrations of restrained nanotubes using modified couple stress theory,” Microsystem Technologies, vol. 24, no. 8, pp. 3425–3435, Aug. 2018, doi: 10.1007/s00542-018-3735-3.
6. S. K. Park and X.-L. Gao, “Bernoulli–Euler beam model based on a modified couple stress theory,” Journal of Micromechanics and Microengineering, vol. 16, no. 11, pp. 2355–2359, Nov. 2006, doi: 10.1088/0960-1317/16/11/015.
7. S. C. Polat and S. M. Bağdatlı, “Investigation of stepped microbeam vibration motions according to modified couple stress theory,” Zeitschrift für Naturforschung A, vol. 78, no. 5, pp. 379–393, May 2023, doi: 10.1515/zna-2022-0286.
8. B. Uzun and M. Ö. Yaylı, “Porosity dependent torsional vibrations of restrained FG nanotubes using modified couple stress theory,” Mater Today Commun, vol. 32, p. 103969, Aug. 2022, doi: 10.1016/j.mtcomm.2022.103969.

9. B. Uzun and M. Ö. Yaylı, “A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory,” International Journal of Engineering and Applied Sciences, vol. 14, no. 1, pp. 1–14, Sep. 2022, doi: 10.24107/ijeas.1064690.
10. B. Uzun and M. Ö. Yaylı, “A Unified Technique for Stability Analysis of an Embedded FG Porous Nano/Microbeam Via Modified Couple Stress Theory,” Nano, Jul. 2024, doi: 10.1142/S1793292024500723.
11. B. Akgöz and Ö. Civalek, “Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams,” Int J Eng Sci, vol. 49, no. 11, pp. 1268–1280, Nov. 2011, doi: 10.1016/j.ijengsci.2010.12.009.
12. B. Uzun, U. Kafkas, and M. Ö. Yaylı, “Free vibration analysis of nanotube based sensors including rotary inertia based on the Rayleigh beam and modified couple stress theories,” Microsystem Technologies, vol. 27, no. 5, pp. 1913–1923, May 2021, doi: 10.1007/s00542-020-04961-z.
13. S. C. Pradhan and G. K. Reddy, “Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM,” Comput Mater Sci, vol. 50, no. 3, pp. 1052–1056, Jan. 2011, doi: 10.1016/J.COMMATSCI.2010.11.001.
14. B. Uzun, Ö. Civalek, and M. Ö. Yaylı, “Vibration of FG nano-sized beams embedded in Winkler elastic foundation and with various boundary conditions,” Mechanics Based Design of Structures and Machines, vol. 51, no. 1, pp. 481–500, Jan. 2023, doi: 10.1080/15397734.2020.1846560.
15. N. Togun and S. Bağdatlı, “Nonlinear Vibration of a Nanobeam on a Pasternak Elastic Foundation Based on Non-Local Euler-Bernoulli Beam Theory,” Mathematical and Computational Applications, vol. 21, no. 1, p. 3, Mar. 2016, doi: 10.3390/mca21010003.
16. B. Uzun and M. Ö. Yaylı, “Porosity and Deformable Boundary Effects on the Dynamic of Nonlocal Sigmoid and Power-Law FG Nanobeams Embedded in the Winkler–Pasternak Medium,” Journal of Vibration Engineering & Technologies, vol. 12, no. 3, pp. 3193–3212, Mar. 2024, doi: 10.1007/s42417-023-01039-8.
17. W.-T. Park, S.-C. Han, W.-Y. Jung, and W.-H. Lee, “Dynamic instability analysis for S-FGM plates embedded in Pasternak elastic medium using the modified couple stress theory,” Steel and Composite Structures, vol. 22, no. 6, pp. 1239–1259, Dec. 2016, doi: 10.12989/scs.2016.22.6.1239.
18. N. Togun and S. M. Bağdatli, “The vibration of nanobeam resting on elastic foundation using modified couple stress theory,” Tehnički glasnik, vol. 12, no. 4, pp. 221–225, Dec. 2018, doi: 10.31803/tg-20180214212115.
19. B. Akgöz and Ö. Civalek, “Free vibration analysis for single-layered graphene sheets in an elastic matrix via modified couple stress theory,” Mater Des, vol. 42, pp. 164–171, Dec. 2012, doi: 10.1016/j.matdes.2012.06.002.
20. B. Akgöz and Ö. Civalek, “Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory,” Meccanica, vol. 48, no. 4, pp. 863–873, May 2013, doi: 10.1007/s11012-012-9639-x.
21. M. Şimşek, “Nonlinear static and free vibration analysis of microbeams based on the nonlinear elastic foundation using modified couple stress theory and He’s variational method,” Compos Struct, vol. 112, pp. 264–272, Jun. 2014, doi: 10.1016/j.compstruct.2014.02.010.
22. U. Kafkas, Y. Ünal, M. Ö. Yaylı, and B. Uzun, “Buckling analysis of perforated nano/microbeams with deformable boundary conditions via nonlocal strain gradient elasticity,” Adv Nano Res, vol. 15, no. 4, pp. 339–353, 2023, doi: 10.12989/anr.2023.15.4.339.
23. L. Luschi and F. Pieri, “An analytical model for the determination of resonance frequencies of perforated beams,” Journal of Micromechanics and Microengineering, vol. 24, no. 5, p. 055004, May 2014, doi: 10.1088/0960-1317/24/5/055004.
24. U. Kafkas, “On the free vibration of a perforated Rayleigh beam with deformable ends,” Engineering Science and Technology, an International Journal, vol. 56, p. 101787, Aug. 2024, doi: 10.1016/J.JESTCH.2024.101787.
25. A. A. Abdelrahman, M. A. Eltaher, A. M. Kabeel, A. M. Abdraboh, and A. A. Hendi, “Free and forced analysis of perforated beams,” Steel and Composite Structures, vol. 31, no. 5, p. 489, 2019.
26. B. Uzun and M. Ö. Yaylı, “Bending Analysis of A Perforated Microbeam With Laplace Transform,” Konya Journal of Engineering Sciences, vol. 11, pp. 23–31, Dec. 2023, doi: 10.36306/konjes.1384835.
27. A. Assie, Ş. D. Akbaş, A. H. Bashiri, A. A. Abdelrahman, and M. A. Eltaher, “Vibration response of perforated thick beam under moving load,” The European Physical Journal Plus, vol. 136, no. 3, p. 283, Mar. 2021, doi: 10.1140/epjp/s13360-021-01224-2.
28. M. A. Eltaher, R. A. Shanab, and N. A. Mohamed, “Analytical solution of free vibration of viscoelastic perforated nanobeam,” Archive of Applied Mechanics, vol. 93, no. 1, pp. 221–243, Jan. 2023, doi: 10.1007/s00419-022-02184-4.
29. U. Kafkas, B. Uzun, M. Ö. Yaylı, and G. Güçlü, “Thermal vibration of perforated nanobeams with deformable boundary conditions via nonlocal strain gradient theory,” Zeitschrift für Naturforschung A, vol. 0, no. 0, Jun. 2023, doi: 10.1515/zna-2023-0088.
30. Ö. Civalek, B. Uzun, and M. Ö. Yaylı, “Size-dependent nonlinear stability response of perforated nano/microbeams via Fourier series,” Archive of Applied Mechanics, vol. 93, no. 12, pp. 4425–4443, Dec. 2023, doi: 10.1007/s00419-023-02501-5.
31. B. Uzun, Ö. Civalek, and M. Ö. Yaylı, “Critical buckling loads of embedded perforated microbeams with arbitrary boundary conditions via an efficient solution method,” Zeitschrift für Naturforschung A, vol. 78, no. 2, pp. 195–207, Feb. 2023, doi: 10.1515/zna-2022-0230.
32. T. Küpeli, Y. H. Çavuş, B. Uzun, and M. Ö. Yaylı, “Free Vibration Response of a Steel Liquid Storage Tank with Porous and Perforated Columns via an Exact Continuum Method,” Gazi University Journal of Science, vol. 36, no. 2, pp. 555–571, Jun. 2023, doi: 10.35378/gujs.1047479.
33. M. A. Koç, M. Eroğlu, and İ. Esen, “Dynamic analysis of high-speed train moving on perforated Timoshenko and Euler–Bernoulli beams,” International Journal of Mechanics and Materials in Design, vol. 18, no. 4, pp. 893–917, Dec. 2022, doi: 10.1007/s10999-022-09610-z.
34. A. A. Abdelrahman, I. Esen, C. Özarpa, and M. A. Eltaher, “Dynamics of perforated nanobeams subject to moving mass using the nonlocal strain gradient theory,” Appl Math Model, vol. 96, pp. 215–235, Aug. 2021, doi: 10.1016/j.apm.2021.03.008.
35. A. A. Abdelrahman, I. Esen, C. Özarpa, R. Shaltout, M. A. Eltaher, and A. E. Assie, “Dynamics of perforated higher order nanobeams subject to moving load using the nonlocal strain gradient theory,” Smart Struct Syst, vol. 28, no. 4, pp. 515–53, Oct. 2021, doi: 10.12989/sss.2021.28.4.515.
36. A. A. Abdelrahman, H. E. Abdel-Mottaleb, A. Aljabri, E. R. I. Mahmoud, and M. A. Eltaher, “Modeling of size dependent buckling behavior of piezoelectric sandwich perforated nanobeams rested on elastic foundation with flexoelectricity,” Mechanics Based Design of Structures and Machines, pp. 1–27, Jun. 2024, doi: 10.1080/15397734.2024.2365918.
37. K. H. Almitani, A. A. Abdelrahman, and M. A. Eltaher, “Stability of perforated nanobeams incorporating surface energy effects,” Steel and Composite Structures, An International Journal, vol. 35, no. 4, pp. 555–566, 2020.
38. M. A. Eltaher, A. M. Kabeel, K. H. Almitani, and A. M. Abdraboh, “Static bending and buckling of perforated nonlocal size-dependent nanobeams,” Microsystem Technologies, vol. 24, no. 12, pp. 4881–4893, Dec. 2018, doi: 10.1007/s00542-018-3905-3.
39. I. Esen, A. A. Abdelrahman, and M. A. Eltaher, “Dynamics analysis of timoshenko perforated microbeams under moving loads,” Eng Comput, vol. 38, no. 3, pp. 2413–2429, Jun. 2022, doi: 10.1007/s00366-020-01212-7.
40. M. A. Eltaher, A. M. Abdraboh, and K. H. Almitani, “Resonance frequencies of size dependent perforated nonlocal nanobeam,” Microsystem Technologies, vol. 24, no. 9, pp. 3925–3937, Sep. 2018, doi: 10.1007/s00542-018-3910-6.
41. K. Mercan, H. M. Numanoglu, B. Akgöz, C. Demir, and Ö. Civalek, “Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix,” Archive of Applied Mechanics, vol. 87, no. 11, pp. 1797–1814, Nov. 2017, doi: 10.1007/s00419-017-1288-z.
42. Ö. Civalek, H. Numanoğlu, and K. Mercan, “Finite Element Model and Size Dependent Stability Analysis of Boron Nitride and Silicon Carbide Nanowires/Nanotubes,” Scientia Iranica, vol. 0, no. 0, pp. 0–0, Apr. 2019, doi: 10.24200/sci.2019.52517.2754.
43. Ç. Demir and Ö. Civalek, “A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix,” Compos Struct, vol. 168, pp. 872–884, May 2017, doi: 10.1016/J.COMPSTRUCT.2017.02.091.
44. J. N. Reddy, Energy Principles and Variational Methods in Applied Mechanics , 2. New York: John Wiley and Sons, 2002.
45. J. K. Phadikar and S. C. Pradhan, “Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates,” Comput Mater Sci, vol. 49, no. 3, pp. 492–499, Sep. 2010, doi: 10.1016/j.commatsci.2010.05.040.
46. M. Şimşek, “Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory,” Compos Struct, vol. 224, p. 111041, Sep. 2019, doi: 10.1016/j.compstruct.2019.111041.
47. S. K. Park and X.-L. Gao, “Variational formulation of a modified couple stress theory and its application to a simple shear problem”, Zeitschrift für angewandte Mathematik und Physik, 59:904–917, 2008, doi: 10.1007/s00033-006-6073-8.
48. J. N. Reddy, “Microstructure-dependent couple stress theories of functionally graded beams”, J Mech Phys Solids, 59:2382–2399, 2011, doi: 10.1016/j.jmps.2011.06.008.