Thermoelastic Vibrations of Nonlocal Nanobeams Resting on a ‎Pasternak Foundation via DPL Model

Document Type : Research Paper


1 Department of Mathematics, College of Science and Arts, Jouf University, Gurayat, Saudi Arabia‎

2 Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt

3 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt


The present work introduces the thermoelastic vibrations of nonlocal nanobeams resting on a two-parameter foundation. The governing equations are formulated for linear Winkler–Pasternak foundation type based on the generalized dual-phase-lag heat conduction and nonlocal beams theories. The nanobeam is subjected to a temperature ramping function. The coupled equations of the problem are formulated and solved by Laplace transform technique. The effects of the nonlocal parameter and different foundation parameters on the field variables are illustrated graphically and discussed. The results obtained are consistent with previous analytical and numerical results.


Main Subjects

‎[1]‎ Eringen, A. C., Nonlocal polar elastic continua, International Journal of Engineering Science, 10(1), 1972, 1–16.‎
‎[2]‎ Eringen, A. C., Edelen, D. G. B., On nonlocal elasticity, International Journal of Engineering Science, 10(3), 1972, 233–248.‎
‎[3]‎ Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, ‎‎54, 1983, 4703–4710.‎
‎[4]‎ Zenkour, A. M., Abouelregal, A. E., The effect of two temperatures on a functionally graded nanobeam induced by a sinusoidal pulse heating, ‎Structural Engineering and Mechanics: an International Journal, 51, 2014, 199 214.‎
‎[5]‎ Zenkour, A. M., Abouelregal, A. E., Effect of harmonically varying heat on FG nanobeams in the context of a nonlocal two-temperature ‎thermoelasticity theory, European Journal of Computational Mechanics, 23, 2014, 1 14.‎
‎[6]‎ Zenkour, A. M., Abouelregal, A. E., Alnefaie, K. A., Abu-Hamdeh, N. H., Aifantis, E. C., Arefined nonlocal thermoelasticity theory for the ‎vibration of nanobeams induced by ramp-type heating, Applied Mathematics and Computation, 248, 2014, 169 183.‎
‎[7]‎ Zenkour, A. M., Abouelregal, A. E., Vibration of FG nanobeams induced by sinusoidal pulse heating via a nonlocal thermoelastic model, Acta ‎Mechanica, 225(12), 2014, 3409–3421.‎
‎[8]‎ Zenkour, A. M., Abouelregal, A. E., Nonlocal thermoelastic nanobeam subjected to a sinusoidal pulse heating and temperature-dependent ‎physical properties, Microsystem Technologies, 21(8), 2015, 1767–1776.‎
‎[9]‎ Pradhan, S.C., Phadikar, J.K., Nonlocal elasticity theory for vibration of nanoplates, Journal of Sound and Vibration, 325(1), 2009, 206-223.‎
‎[10]‎ Kolahchi, R., Bidgoli, A.M., Size-dependent sinusoidal beam model for dynamic instability of single-walled carbon nanotubes, Applied ‎Mathematics and Mechanics, 37, 2016, 265-274.‎
‎[11]‎ Besseghier, A., Houari, M.S.A., Tounsi, A., Mahmoud, S.R., Free vibration analysis of embedded nanosize FG plates using a new nonlocal ‎trigonometric shear deformation theory, Smart Structures and Systems, 19(6), 2017, 601-614.‎
‎[12]‎ Mouffoki, A., Adda Bedia, E.A., Houari, M.S.A., Tounsi, A., Mahmoud, S.R., Vibration analysis of nonlocal advanced nanobeams in hygro-‎thermal environment using a new two-unknown trigonometric shear deformation beam theory, Smart Structures and Systems, 20(3), 2017, 369-‎‎383.‎
‎[13]‎ Norouzzadeh, A., Ansari, R., Finite element analysis of nano-scale Timoshenko beams using the integral model of nonlocal elasticity, Physica E: ‎Low-dimensional Systems and Nanostructures, 88, 2017, 194–200.‎
‎[14]‎ Aria, A. I., Friswell, M. I., Rabczuk, T., Thermal vibration analysis of cracked nanobeams embedded in an elastic matrix using finite element ‎analysis, Composite Structures, 212, 2019, 118-128.‎
‎[15]‎ Sedighi, H. M., Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity ‎theory, Acta Astronautica, 95, 2014, 111–123.‎
‎[16]‎ Sedighi, H. M., Daneshmand, F., Static and dynamic pull-in instability of multi-walled carbon nanotube probes by He’s iteration perturbation ‎method, Journal of Mechanical Science and Technology, 28(9), 2014, 3459–3469. ‎
‎[17]‎ He, C.-H., He, J.-H., Sedighi, H. M., Fangzhu (方诸): An ancient Chinese nanotechnology for water collection from air: History, mathematical ‎insight, promises, and challenges, Mathematical Methods in the Applied Sciences, 2020,‎
‎[18]‎ Ouakad, H. M., Sedighi, H. M., Static response and free vibration of MEMS arches assuming out-of-plane actuation pattern, International Journal ‎of Non-Linear Mechanics, 110, 2019, 44–57.‎
‎[19]‎ Sedighi, H. M., Ouakad, H. M., Dimitri, R., Tornabene, F., Stress-driven nonlocal elasticity for the instability analysis of fluid-conveying C-BN ‎hybrid-nanotube in a ‎magneto-thermal environment, Physica Scripta, 2020, ‎
‎[20]‎ Sedighi, H. M., Divergence and flutter instability of magneto-thermo-elastic C-BN hetero-nanotubes conveying fluid, Acta Mechanica Sinica, 36, ‎‎2020, 381-396.‎
‎[21]‎ Shariati, A., Jung, D. W., Sedighi, H. M., Żur, K. K., Habibi, M., Safa, M., On the Vibrations and Stability of Moving Viscoelastic Axially ‎Functionally Graded Nanobeams, Materials, 2020,‎
‎[22]‎ Salamat, D., Sedighi, H. M., The effect of small scale on the vibrational behavior of single-walled carbon nanotubes with a moving ‎nanoparticle, ‎Journal of Applied and Computational Mechanics, 3(3), 2017, 208-217.‎
‎[23]‎ SoltanRezaee, M., Bodaghi , M., Nonlinear dynamic stability of piezoelectric thermoelastic electromechanical resonators, Scientific Reports, (10), ‎‎2020‎
‎[24]‎ Sari, M. S., Al-Kouz, W.G., Vibration analysis of non-uniform orthotropic Kirchhoff plates resting on elastic foundation based on nonlocal ‎elasticity theory, International Journal of Mechanical Sciences , 114, 2016, 1-11.‎
‎[25]‎ Ragb, O., Mohamed, M., Matbuly, M.S., Free vibration of a piezoelectric nanobeam resting on nonlinear Winkler-Pasternak foundation by ‎quadrature methods, Heliyon, 5(6), 2019, e01856.‎
‎[26]‎ Niknam, H., Aghdam, M., A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic ‎foundation, Composite Structures, 119, 2015, 452-462.‎
‎[27]‎ Ghanvanloo, E., Daneshmand, F., Rafiei, M., Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear ‎viscoelastic Winkler foundation, Physica E: Low-dimensional Systems and Nanostructures, 42(9), 2010, 2218–2224.‎
‎[28]‎ Kiani, K., Vibration analysis of elastically restrained double-walled carbon nanotubes on elastic foundation subject to axial load using nonlocal ‎shear deformable beam theories, International Journal of Mechanical Sciences, 68, 2013, 16–34.‎
‎[29]‎ Zenkour, A., Ebrahimi, F., Barati, M. R., Buckling analysis of a size-dependent functionally graded nanobeam resting on Pasternak's ‎foundations, International Journal of Nano Dimension, 10 (2), 2019, 141-153.‎
‎[30]‎ Zenkour, A. M., Sobhy, M., Nonlocal elasticity theory for thermal buckling of nanoplates lying on Winkler-Pasternak elastic substrate medium, ‎Physica E: Low-dimensional Systems and Nanostructures, 53, 2013, 251–259.‎
‎[31]‎ Tzou, D. Y., A unified approach for heat conduction from macro- to micro-scales. Journal of Heat Transfer, 117(1), 1995, 8-16. ‎
‎[32]‎ Tzou, D. Y., Macro-to-Microscale Heat Transfer: The Lagging Behavior, Washington, DC, Taylor & Francis, 1996. ‎
‎[33]‎ Tzou, D. Y., Experimental support for the Lagging behavior in heat propagation, Journal of Thermophysics and Heat Transfer, 9(4), 1995, 686-693.‎
‎[34]‎ Kirs, M., Eerme, M., Bassir, D., Tungel, E., Application of HOHWM for Vibration Analysis of Nanobeams, Key Engineering Materials, 799, 2019, 230-‎‎235.‎
‎[35]‎ Hetényi, M., A general solution for the bending of beams on an elastic foundation of arbitrary continuity, Journal of Applied Physics, 21, 1950, 55–‎‎58.‎
‎[36]‎ Refaeinejad, V., Rahmani, O., Hosseini, S.A., An analytical solution for bending, buckling, and free vibration of FG nanobeam lying on Winkler-‎Pasternak elastic foundation using different nonlocal higher order shear deformation beam theories, Scientia Iranica, 24(3), 2017, 1635-1653.‎
‎[37]‎ Zenkour, A. M., Free vibration of a microbeam resting on Pasternak’s foundation via the Green–Naghdi thermoelasticity theory without energy ‎dissipation, Journal of Low Frequency Noise, Vibration and Active Control, 35(4), 2016, 303–311.‎
‎[38]‎ Abouelregal, A.E., Zenkour, A.M., Vibration of FG viscoelastic nanobeams due to a periodic heat flux via fractional derivative model, Journal of ‎Computational Applied Mechanics, 50(1), 2019, 148-156. ‎
‎[39]‎ Abouelregal, A. E., Zenkour, A. M., Nonlocal thermoelastic model for temperature-dependent thermal conductivity nanobeams due to dynamic ‎varying loads, Microsystem Technologies, 24(2), 2018, 1189–1199.‎
‎[40]‎ Bensaid, I., Bekhadda, A., Kerboua, B., Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on ‎nonlocal strain gradient theory, Advances in Nano Research, 6(3), 2018, 279-298.‎
‎[41]‎ Yokoyama, T., Vibrations and transient responses of Timoshenko beams resting on elastic foundations, Archive of Applied Mechanics, 57, 1987, ‎‎81–90.‎
‎[42]‎ Togun, N., Bagdatli, S.M., Nonlinear vibration of a nanobeam on Pasternak elastic foundation based on nonlocal Euler-Bernoulli beam theory, ‎Mathematical and Computational Applications, 21(1), 2016, 1–19.‎
‎[43]‎ Mashat, D.S., Zenkour, A.M., Abouelregal, A.E., Thermoviscoelastic Vibrations of a Micro-Scale Beam Subjected to Sinusoidal Pulse Heating, ‎International Journal of Acoustics and Vibration, 22(2), 2017, 260 – 269.‎
‎[44]‎ Abouelregal, A.E., Mohammed, W.W., Effects of nonlocal thermoelasticity on nanoscale beams based on couple stress theory, Mathematical ‎Methods in the Applied Sciences, 2020,‎
‎[45]‎ Abouelregal A.E., Marin, M., The response of nanobeams with temperature-dependent properties using state-space method via modified ‎couple stress theory, Symmetry, 2020,‎