Advances in nano research

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Bishop theory and longitudinal vibration of nano-beams by two-phase local/nonlocal elasticity

  • Received : 2022.01.31
  • Accepted : 2023.01.17
  • Published : 2023.07.25
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Abstract

In this paper, Bishop theory performs longitudinal vibration analysis of Nano-beams. Its governing equation, due to integrated displacement field and more considered primarily effects compared with other theories, enjoys fully completed status, and more reliable results as well. This article aims to find how Bishop theory and Two-phase elasticity work together. In other words, whether Bishop theory will be compatible with Two-phase local/nonlocal elasticity. Hamilton's principle is employed to derive governing equation of motion, and then the 6th order of Generalized Differential Quadrature Method (GDQM) as a constructive numerical method is utilized to attain the discretized two-phase formulation. To acquire a proper verification procedure, exact solution is prepared to be compared with current results. Furthermore, the effects of key parameters on the objective are investigated.

Keywords

References

  1. Apuzzo, A., Barretta, R., Luciano, R., De Sciarra, F.M. and Penna, R. (2017), "Free vibrations of Bernoulli-Euler nano-beams by the stress-driven nonlocal integral model". Compos. Part B Eng., 123, 105-111, https://doi.org/10.1016/j.compositesb.2017.03.057
  2. Bergman, R.M. (1968), "Asymptotic analysis of some plane problems of the theory op elasticity with couple stresses". J. Appl. Math. Mech., 32(6), 1085-1090. https://doi.org/10.1016/0021-8928(68)90035-X
  3. Bert, C.W. and Malik, M. (1996), "Differential quadrature method in computational mechanics: A review", ASME. Appl. Mech. Rev., 49(1), 1-28. https://doi.org/10.1115/1.3101882
  4. Besseghier, A., Heireche, H., Bousahla, A.A., Tounsi, A. and Benzair, A. (2015), "Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix", Adv. Nano Res., 3(1), 029. https://doi.org/10.12989/anr.2015.3.1.029
  5. Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Beg, O.A. and Mahmoud, S.R. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425
  6. Challamel, N., Zhang, Z., Wang, C.M., Reddy, J.N., Wang, Q., Michelitsch, T. and Collet, B. (2014), "On nonconservativeness of Eringen's nonlocal elasticity in beam mechanics: correction from a discrete-based approach", Arch. Appl. Mech., 84(9), 1275-1292. https://doi.org/10.1007/s00419-014-0862-x
  7. Civalek, O . and Numanoglu, H.M. (2020), "Nonlocal finite element analysis for axial vibration of embedded love-bishop nanorods", Int. J. Mech. Sci., 188, 105939. https://doi.org/10.1016/j.ijmecsci.2020.105939
  8. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Applied Math. Comput., 218(14), 7406-7420. https://doi.org/10.1016/j.amc.2011.12.090
  9. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  10. Eringen, A.C. (1984), "Theory of nonlocal elasticity and some applications", Res. Mech., 21(4), 313-342. https://doi.org/10.21236/ada145201
  11. Eringen, A.C. and Wegner, J.L. (2003), "Nonlocal continuum field theories", Appl. Mech. Rev., 56(2), B20-B22. https://doi.org/10.1115/1.1553434
  12. Fernandez-Saez, J., Zaera, R., Loya, J.A. and Reddy, J. (2016), "Bending of Euler-Bernoulli beams using Eringen's integral formulation: a paradox resolved", Int. J. Eng. Sci., 99, 107-116. https://doi.org/10.1016/j.ijengsci.2015.10.013
  13. Fleck, N.A., Muller, G.M., Ashby, M.F. and Hutchinson, J.W. (1994), "Strain gradient plasticity: theory and experiment", Acta Metallurgica et Materialia, 42(2), 475-487. https://doi.org/10.1016/0956-7151(94)90502-9
  14. Hosseini, S.A., Khosravi, F. and Ghadiri, M. (2020), "Moving axial load on dynamic response of single-walled carbon nanotubes using classical, Rayleigh and Bishop rod models based on Eringen's theory", J. Vib. Control, 26(11-12), 913-928. https://doi.org/10.1177/1077546319890170
  15. Hsu, J.C., Lee, H.L. and Chang, W.J. (2011), "Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory", Curr. Appl. Phys., 11(6), 1384-1388. https://doi.org/10.1016/j.cap.2011.04.026
  16. Jalaei, M.H., Thai, H.T. and Civalek, Ӧ. (2022), "On viscoelastic transient response of magnetically imperfect functionally graded nanobeams", Int. J. Eng. Sci., 172, 103629. https://doi.org/10.1016/j.ijengsci.2022.103629
  17. Khaniki, H.B. (2018), "On vibrations of nanobeam systems", Int. J. Eng. Sci., 124, 85-103. https://doi.org/10.1016/j.ijengsci.2017.12.010
  18. Lam, D.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  19. Lei, Y., Adhikari, S. and Friswell, M.I. (2013), "Vibration of nonlocal Kelvin-Voigt viscoelastic damped Timoshenko beams", Int. J. Eng. Sci., 66, 1-13. https://doi.org/10.1016/j.ijengsci.2013.02.004
  20. Li, H.B. and Wang, X. (2012), "Effect of small scale on the dynamic characteristic of carbon nanotubes under axially oscillating loading", Physica E, 46, 198-205. https://doi.org/10.1016/j.physe.2012.09.015
  21. Lim, C.W., Zhang, G. and Reddy, J. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001
  22. Lin, F., Tong, L.H., Shen, H.S., Lim, C.W. and Xiang, Y. (2020), "Assessment of first and third order shear deformation beam theories for the buckling and vibration analysis of nanobeams incorporating surface stress effects", Int. J. Mech. Sci., 186, 105873. https://doi.org/10.1016/j.ijmecsci.2020.105873
  23. Lu, L., Guo, X. and Zhao, J. (2017), "A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms". Int. J. Eng. Sci., 119, 265-277. https://doi.org/10.1016/j.ijengsci.2017.06.024
  24. Mashat, D.S., Zenkour, A.M. and Sobhy, M. (2016), "Investigation of vibration and thermal buckling of nanobeams embedded in an elastic medium under various boundary conditions", J. Mech., 32(3), 277-287. https://doi.org/10.1017/jmech.2015.83
  25. Mirafzal, A. and Fereidoon, A. (2017), "Dynamic characteristics of temperature-dependent viscoelastic FG nanobeams subjected to 2D-magnetic field under periodic loading", Appl. Phys. A, 123(4), 1-13. https://doi.org/10.1007/s00339-017-0829-1
  26. Naderi, A., Behdad, S., Fakher, M. and Hosseini-Hashemi, S. (2020), "Vibration analysis of mass nanosensors with considering the axial-flexural coupling based on the two-phase local/nonlocal elasticity", Mech. Syst. Signal Pr., 145, 106931. https://doi.org/10.1016/j.ymssp.2020.106931
  27. Nazemnezhad, R. and Kamali, K. (2018), "Free axial vibration analysis of axially functionally graded thick nanorods using nonlocal Bishop's theory", Steel Compos. Struct., 28(6), 749-758. https://doi.org/10.12989/scs.2018.28.6.749
  28. Nazemnezhad, R. and Shokrollahi, H. (2022), "Axial frequency analysis of axially functionally graded Love-Bishop nanorods using surface elasticity theory", Steel Compos. Struct., 42(5), 699-710. https://doi.org/10.12989/scs.2022.42.5.699
  29. Numanoglu, H.M. and Civalek, O . (2019), "On the torsional vibration of nanorods surrounded by elastic matrix via nonlocal FEM", Int. J. Mech. Sci., 161, 105076. https://doi.org/10.1016/j.ijmecsci.2019.105076
  30. Numanoglu, H.M. and Civalek, O . (2022), "Novel size-dependent finite element formulation for modal analysis of cracked nanorods", Materials Today Commun., 31, 103545. https://doi.org/10.1016/j.mtcomm.2022.103545
  31. Numanoglu, H.M., Akgoz, B. and Civalek, O . (2018), "On dynamic analysis of nanorods", Int. J. Eng. Sci., 130, 33-50. https://doi.org/10.1016/j.ijengsci.2018.05.001
  32. Numanoglu, H.M., Ersoy, H., Akgoz, B. and Civalek, O . (2022), "A new eigenvalue problem solver for thermo-mechanical vibration of Timoshenko nanobeams by an innovative nonlocal finite element method", Math. Method Appl. Sci., 45(5), 2592-2614. https://doi.org/10.1002/mma.7942
  33. Polyanin, A.D. and Manzhirov, A.V. (2008), Handbook of Integral Equations, Chapman and Hall/CRC.
  34. Quan, J.R. and Chang, C.T. (1989), "New insights in solving distributed system equations by the quadrature method-I. Analysis", Comput. Chem. Eng., 13(7), 779-788. https://doi.org/10.1016/0098-1354(89)85051-3
  35. Rao, S.S. (2019), Vibration of Continuous Systems, John Wiley & Sons.
  36. Reddy, J. (2007), "Nonlocal theories for bending, buckling and vibration of beams", Int. J. Eng. Sci., 45(2-8), 288-307. https://doi.org/10.1016/j.ijengsci.2007.04.004
  37. Shafiei, N., Mirjavadi, S.S., Afshari, B.M., Rabby, S. and Hamouda, A.M.S. (2017), "Nonlinear thermal buckling of axially functionally graded micro and nanobeams", Comp. Struct., 168, 428-439. https://doi.org/10.1016/j.compstruct.2017.02.048
  38. Shen, Y., Chen, Y. and Li, L. (2016), "Torsion of a functionally graded material", Int. J. Eng. Sci., 109, 14-28. https://doi.org/10.1016/j.ijengsci.2016.09.003
  39. Shu, C. and Richards, B.E. (1992), "Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations", Int. J. Numer. Method Fluids, 15(7), 791-798. https://doi.org/10.1002/fld.1650150704
  40. Simsek, M. and Yurtcu, H.H. (2013), "Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory", Comp. Struct., 97, 378-386. https://doi.org/10.1016/j.compstruct.2012.10.038
  41. Thai, H.T. (2012), "A nonlocal beam theory for bending, buckling, and vibration of nanobeams", Int. J. Eng. Sci., 52, 56-64. https://doi.org/10.1016/j.ijengsci.2011.11.011
  42. Wang, Y., Zhao, Y.B. and Wei, G.W. (2003), "A note on the numerical solution of high-order differential equations", J. Comput. Appl. Math., 159(2), 387-398. https://doi.org/10.1016/S0377-0427(03)00541-7
  43. Wang, Y.B., Zhu, X.W. and Dai, H.H. (2016), "Exact solutions for the static bending of Euler-Bernoulli beams using Eringen's two-phase local/nonlocal model", Aip Adv., 6(8), 085114. https://doi.org/10.1063/1.4961695
  44. Wu, T.Y. and Liu, G.R. (2000), "Application of generalized differential quadrature rule to sixth-order differential equations", Commun. Numer. Methods Eng., 16(11), 777-784. https://doi.org/10.1002/10990887(200011)16:11<777::AIDCNM375>3.0.CO;2-6
  45. Wu, T.Y. and Liu, G.R. (2001), "Free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule", Int. J. Solids Struct., 38(44-45), 7967-7980. https://doi.org/10.1016/S0020-7683(01)00077-4
  46. Zhang, G.Y. and Gao, X.L. (2020), "A new Bernoulli-Euler beam model based on a reformulated strain gradient elasticity theory", Math. Mech. Solids, 25(3), 630-643. https://doi.org/10.1177/1081286519886003
  47. Zhang, G.Y., Gao, X.L., Zheng, C.Y. and Mi, C.W. (2021), "A non-classical Bernoulli-Euler beam model based on a simplified micromorphic elasticity theory", Mech. Mater., 161, 103967. https://doi.org/10.1016/j.mechmat.2021.103967
  48. Zhao, X., Zheng, S. and Li, Z. (2020), "Effects of porosity and flexoelectricity on static bending and free vibration of AFG piezoelectric nanobeams", Thin Walled Struct., 151, 106754. https://doi.org/10.1016/j.tws.2020.106754
  49. Zhu, X. and Li, L. (2017), "Twisting statics of functionally graded nanotubes using Eringen's nonlocal integral model", Compos. Struct., 178, 87-96. https://doi.org/10.1016/j.compstruct.2017.06.067
  50. Zhu, X., Wang, Y. and Dai, H.H. (2017), "Buckling analysis of Euler-Bernoulli beams using Eringen's two-phase nonlocal model", Int. J. Eng. Sci., 116, 130-140. https://doi.org/10.1016/j.ijengsci.2017.03.008