Size-Dependent Bending Response of Perforated Nanobeams on Winkler-Pasternak Foundation

Year 2025, Volume: 17 Issue: 1, 1 - 16, 01.05.2025

Uğur Kafkas

https://doi.org/10.24107/ijeas.1590000

Cited By: 1

Abstract

This study investigates the bending response of perforated nanobeams resting on Winkler-Pasternak elastic foundation (WPEF), using Eringen's theory of nonlocal elasticity (ENET). The analysis examines how various parameters affect the mechanical response of the nanobeam, including the nonlocal parameter, foundation parameters, filling ratio, and number of holes. Results indicate that an increase in the nonlocal parameter produces larger transverse displacements compared to classical beam theory, while the stiffness decreases due to nanoscale effects. The elastic foundation parameters significantly influence beam behavior, with the Pasternak model proving more effective than the Winkler model (WEF) in reducing displacement. Analysis of hole properties reveals that higher filling ratios increase beam stiffness, while an increase in the number of holes decreases nanobeam stiffness. These findings are crucial for optimizing the design of nanoelectromechanical systems and other nanostructured devices where bending behavior affects performance.

Keywords

Nonlocal Elasticity , Perforated Nanobeam , Winkler-Pasternak Foundation , Filling Ratio

References

• Eltaher, M.A., Khairy, A., Sadoun, A.M., Omar, F.A., Static and buckling analysis of functionally graded Timoshenko nanobeams. Applied Mathematics and Computation, 229, 283–295, 2014.
• Demir, C., Civalek, O., On the analysis of microbeams. International Journal of Engineering Science, 121, 14–33, 2017.
• Yaylı, M.O., Stability analysis of a rotationally restrained microbar embedded in an elastic matrix using strain gradient elasticity. Curved and Layered Structures, 6(1),1–10, 2019.
• Eringen, A.C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703–4710, 1983.
• Mindlin, R.D., Second gradient of strain and surface-tension in linear elasticity. International journal of solids and structures, 1(4), 417–438, 1965.
• Yang, F., Chong, A.C.M., Lam, D.C.C., Tong, P., Couple stress based strain gradient theory for elasticity. Int J Solids Struct, 39(10), 2731–2743, 2002.
• Lim, C.W., Zhang, G., Reddy, J.N., A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, 298–313, 2015.
• Civalek, O., Uzun, B., Yaylı, M.O., Frequency, bending and buckling loads of nanobeams with different cross sections. Advances in nano research, 9(2), 91–104, 2020.
• Uzun, B., Yaylı, M.Ö., A Finite Element Solution for Bending Analysis of a Nanoframe using Modified Couple Stress Theory. International Journal of Engineering and Applied Sciences, 14(1), 1–14, 2022.
• Reddy, J.N., Nonlocal theories for bending, buckling and vibration of beams. International journal of engineering science, 45(2-8), 288–307, 2007.
• Zghal, S., Ataoui, D., Dammak, F., Static bending analysis of beams made of functionally graded porous materials. Mechanics Based Design of Structures and Machines, 50(3), 1012–1029, 2022.
• Kafkas, U., Değiştirilmiş Gerilme Cifti Teorisi ile Gözenekli Fonksiyonel Derecelendirilmiş Konsol Nanokirişlerin Statik Analizi. Uludağ University Journal of The Faculty of Engineering, 29(2), 393–412, 2024
• Numanoglu, H.M., Civalek, O., Elastic Beam Model and Bending Analysis of Silver Nanowires. International Journal of Engineering and Applied Sciences, 10(1), 13–20, 2018.
• Akgoz, B., Civalek, O., Investigation of Size Effects on Static Response of Single-Walled Carbon Nanotubes Based on Strain Gradient Elasticity. International Journal of Computational Methods, 9(02): 1240032, 2012.
• Akgoz, B., Civalek, O., Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment. Composites Part B: Engineering, 150, 68–77, 2018.
• Civalek, O., Uzun, B., Yaylı, M.O., Akgoz, B., Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method. The European Physical Journal Plus, 135(4), 381, 2020.
• Akgoz, B., Civalek, O., Longitudinal vibration analysis for microbars based on strain gradient elasticity theory. Journal of Vibration and Control, 20(4), 606–616, 2014.
• Civalek, O., Numanoglu, H.M., Nonlocal finite element analysis for axial vibration of embedded love–bishop nanorods. International Journal of Mechanical Sciences, 188, 105939, 2020.
• Uzun, B., Yaylı, M.O., A solution method for longitudinal vibrations of functionally graded nanorods. International Journal of Engineering and Applied Sciences, 12(2),78-87, 2020.
• Akbas, S.D., Dastjerdi, S., Akgoz, B., Civalek, O., Dynamic Analysis of Functionally Graded Porous Microbeams under Moving Load. Transport in Porous Media, 142, 209–27, 2022.
• Aydogdu, M., Arda, M., Forced vibration of nanorods using nonlocal elasticity. Advances in Nano Research, 4(4), 265-279, 2016.
• Dastjerdi, S., Malikan, M., Akgoz, B., Civalek. O., Wiczenbach, T., Eremeyev, V.A., On the deformation and frequency analyses of SARS-CoV-2 at nanoscale. International Journal of Engineering Science, 170:103604, 2022.
• Aydogdu, M., Arda, M. Torsional vibration analysis of double walled carbon nanotubes using nonlocal elasticity. International Journal of Mechanics and Materials in Design, 12, 71–84, 2016.
• Yaylı, M.O., An efficient solution method for the longitudinal vibration of nanorods with arbitrary boundary conditions via a hardening nonlocal approach. Journal of Vibration and Control, 24(11), 2230–2246, 2018.
• Abouelregal, A.E., Akgoz, B., Civalek, O., Nonlocal thermoelastic vibration of a solid medium subjected to a pulsed heat flux via Caputo–Fabrizio fractional derivative heat conduction. Applied Physics A, 128(8), 660, 2022.
• Yaylı, M.O., Free vibration analysis of a rotationally restrained (FG) nanotube. Microsystem Technologies, 25, 3723–3734, 2019.
• Yaylı, M.O., Uzun, B., Deliktas, B., Buckling analysis of restrained nanobeams using strain gradient elasticity. Waves in Random and Complex Media, 32(6), 2960-2979, 2021.
• Civalek, O., Uzun, B., Yaylı, M.O., Thermal buckling analysis of a saturated porous thick nanobeam with arbitrary boundary conditions. Journal of Thermal Stresses, 46(1), 1–21, 2023.
• Yayli, M.O., Buckling Analysis of a Rotationally Restrained Single Walled Carbon Nanotube. Acta Phys Pol A, 127(3), 678–683, 2015.
• Civalek, O., Uzun, B., Yaylı, M.O., An effective analytical method for buckling solutions of a restrained FGM nonlocal beam. Computational and Applied Mathematics, 41(2), 67, 2022.
• Yaylı, M.O., Stability analysis of gradient elastic microbeams with arbitrary boundary conditions. Journal of Mechanical Science and Technology, 29, 3373–80, 2015.
• Kafkas, U., Evaluation of Critical Buckling Loads of Short Fiber Reinforced Nanobeams According to Different Beam Theories. Kafkas University Institute of Natural and Applied Science Journal, 17(1), 1–14, 2024.
• Eltaher, M.A., Kabeel, A.M., Almitani, K.H., Abdraboh, A.M., Static bending and buckling of perforated nonlocal size-dependent nanobeams. Microsystem Technologies, 24, 4881–4893, 2018.
• Abdelrahman, A.A., Mohamed, N.A., Eltaher, M.A., Static bending of perforated nanobeams including surface energy and microstructure effects. Eng Comput, 38, 415–435, 2022.
• Rebeiz GM. RF MEMS: Theory, Design, and Technology, Wiley, 2003.
• Bendali, A., Labedan, R., Domingue, F., Nerguizian, V., Holes Effects on RF MEMS Parallel Membranes Capacitors, 2006 Canadian Conference on Electrical and Computer Engineering, IEEE, Ottawa, ON, Canada, 7-10 May 2006.
• Uzun, B., Yaylı, M.O., Bending Analysis of A Perforated Microbeam With Laplace Transform. Konya Journal of Engineering Sciences, 11, 23–31, 2023.
• Abdelrahman, A.A., Abd El Mottaleb, H.E., Eltaher, M.A., On bending analysis of perforated microbeams including the microstructure effects. Structural Engineering and Mechanics, An Int’l Journal, 76(6), 765–779, 2020.
• Abdelrahman, A.A., Saleem, H.A., Abdelhaffez, G.S., Eltaher, M.A., On Bending of Piezoelectrically Layered Perforated Nanobeams Embedded in an Elastic Foundation with Flexoelectricity. Mathematics, 11(5), 1162, 2023.
• Assie, A., Akbas, Ş.D., Bashiri, A.H., Abdelrahman, A.A., Eltaher, M.A., Vibration response of perforated thick beam under moving load. The European Physical Journal Plus, 136, 283, 2021
• Kupeli, T., Çavus, Y.H., Uzun, B., Yayli, M.O., Free Vibration Response of a Steel Liquid Storage Tank with Porous and Perforated Columns via an Exact Continuum Method. Gazi University Journal of Science, 36(2), 555–571, 2023.
• Eltaher, M.A., Shanab, R.A., Mohamed, N.A., Analytical solution of free vibration of viscoelastic perforated nanobeam. Archive of Applied Mechanics, 93(1), 221–243, 2023.
• Eltaher, M.A., Mohamed, N.A., Vibration of nonlocal perforated nanobeams with general boundary conditions. Smart Struct Syst, 25(4), 501–514, 2020.
• Abdelrahman, A.A., Eltaher, M.A., Kabeel, A.M., Abdraboh, A.M., Hendi, A.A., Free and forced analysis of perforated beams. Steel and Composite Structures, 31(5), 489, 2019.
• Kafkas, U., On the free vibration of a perforated Rayleigh beam with deformable ends. Engineering Science and Technology, an International Journal, 56:101787, 2024.
• Eltaher, M.A., Abdraboh, A.M., Almitani, K.H., Resonance frequencies of size dependent perforated nonlocal nanobeam. Microsystem Technologies, 24, 3925–3937, 2018.
• Esen, I., Abdelrahman, A.A., Eltaher, M.A., Dynamics analysis of Timoshenko perforated microbeams under moving loads. Engineering with Computers, 38, 2413–2429, 2022.
• Uzun, B., Civalek, O., Yaylı, M.O., Critical buckling loads of embedded perforated microbeams with arbitrary boundary conditions via an efficient solution method. Zeitschrift für Naturforschung A, 78(2), 195–207, 2023.
• Civalek, O., Uzun, B., Yaylı, M.O., Size-dependent nonlinear stability response of perforated nano/microbeams via Fourier series. Archive of Applied Mechanics, 93(12), 4425–4443, 2023.
• Almitani, K.H., Abdelrahman, A.A., Eltaher, M.A., Stability of perforated nanobeams incorporating surface energy effects. Steel and Composite Structures, An International Journal, 35(4), 555–566, 2020.
• Abdelrahman, A.A., Eltaher, M.A., On bending and buckling responses of perforated nanobeams including surface energy for different beams theories. Engineering with Computers, 38(3), 2385–2411, 2022.
• Togun, N., Bagdatlı, S., Nonlinear Vibration of a Nanobeam on a Pasternak Elastic Foundation Based on Non-Local Euler-Bernoulli Beam Theory. Mathematical and Computational Applications, 21, 3, 2016.
• Uzun, B., Yayli, M.O., 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, 12(3), 3193–3212, 2024.
• Uzun, B., Yaylı, M.O., Winkler‐Pasternak foundation effect on the buckling loads of arbitrarily rigid or restrained supported nonlocal beams made of different FGM and porosity distributions. ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 104(2), e202300569, 2024.
• Amirian, B., Hosseini-Ara, R., Moosavi, H., Thermal vibration analysis of carbon nanotubes embedded in two-parameter elastic foundation based on nonlocal Timoshenko’s beam theory. Archives of Mechanics. 64(6), 581–602, 2013.
• Civalek, O., Demir, C., A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Applied Mathematics and Computation, 289, 335–352, 2016.
• Uzun, B., Civalek, O., Nonlocal FEM Formulation for Vibration Analysis of Nanowires on Elastic Matrix with Different Materials. Mathematical and Computational Applications, 24(2), 38, 2019
• Numanoglu, H.M., Civalek, O., On the torsional vibration of nanorods surrounded by elastic matrix via nonlocal FEM. International Journal of Mechanical Sciences, 161–162, 105076, 2019.
• Uzun, B., Kafkas, U., Yaylı, M.O., Guclu, G., Torsional vibration behavior of a restrained non-circular nanowire in an elastic matrix. Mechanics Based Design of Structures and Machines, 52(10), 1–33, 2024.
• Mercan, K., Numanoglu, H.M., Akgoz, B., Demir, C., Civalek, O., Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix. Archive of Applied Mechanics, 87, 1797–1814, 2017.
• Yayli, M.O., Free vibration analysis of a single‐walled carbon nanotube embedded in an elastic matrix under rotational restraints. Micro Nano Lett, 13(2), 202–206, 2018.
• Uzun, B., Yaylı, M.O., Nonlocal vibration analysis of Ti-6Al-4V/ZrO2 functionally graded nanobeam on elastic matrix. Arabian Journal of Geosciences, 13(4), 155, 2020.
• Yaylı, M.O., On the torsional vibrations of restrained nanotubes embedded in an elastic medium. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40:419, 2018.
• Uzun, B., Yaylı, M.O., Stability analysis of arbitrary restrained nanobeam embedded in an elastic medium via nonlocal strain gradient theory. The Journal of Strain Analysis for Engineering Design, 58(8), 672–83, 2023.
• Kafkas, U., Unal, Y., Yaylı, M.O., Uzun, B., Buckling analysis of perforated nano/microbeams with deformable boundary conditions via nonlocal strain gradient elasticity. Advances in nano research, 15(4), 339–353, 2023.
• Abdelrahman, A.A., Abdel-Mottaleb, H.E., Aljabri, A., Mahmoud, E.R.I., Eltaher, M.A., Modeling of size dependent buckling behavior of piezoelectric sandwich perforated nanobeams rested on elastic foundation with flexoelectricity. Mechanics Based Design of Structures and Machines, 53(1), 1–27, 2024.
• Kafkas, U., Uzun, B., Yaylı, M.O., Guçlu, G., Thermal vibration of perforated nanobeams with deformable boundary conditions via nonlocal strain gradient theory. Zeitschrift für Naturforschung A, 78(8), 681-701, 2023.
• Mohamed, N.A., Shanab, R.A., Eltaher, M.A., Abdelrahman, A.A., Vibration response of viscoelastic perforated higher-order nanobeams rested on an elastic substrate under moving load. Acta Mechanica, 235(2), 1213–1233, 2024.
• Akgoz, B., Civalek, O., Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity. Composite Structures, 134, 294–301, 2015.
• Demir, C., Mercan, K., Numanoglu, H.M., Civalek, O., Bending Response of Nanobeams Resting on Elastic Foundation. Journal of Applied and Computational Mechanics, 4(2), 105–114, 2018.
• Numanoglu, H.M., Dynamic Analysis of Nano Scaled Continuous and Discrete Structures Based on Nonlocal Finite Element Formulation (NL–FEM). Akdeniz University Graduate School of Natural and Applied Sciences, Antalya, Master Thesis, 2019.
• Simsek, M., Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory. Composite Structures, 224, 111041, 2019.
• Luschi, L., Pieri, F., An analytical model for the determination of resonance frequencies of perforated beams. Journal of Micromechanics and Microengineering, 24(5), 055004, 2014.
• Chen, W.Q., Lu, C.F., Bian, Z.G., A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Applied mathematical modelling, 28(10), 877–890, 2004.