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On the vibration of nanobeams with consistent two-phase nonlocal strain gradient theory: exact solution and integral nonlocal finite-element model

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Abstract

Recently, it has been proved that the common nonlocal strain gradient theory has inconsistence behaviors. The order of the differential nonlocal strain gradient governing equations is less than the number of all mandatory boundary conditions, and therefore, there is no solution for these differential equations. Given these, for the first time, transverse vibrations of nanobeams are analyzed within the framework of the two-phase local/nonlocal strain gradient (LNSG) theory, and to this aim, the exact solution as well as finite-element model are presented. To achieve the exact solution, the governing differential equations of LNSG nanobeams are derived by transformation of the basic integral form of the LNSG to its equal differential form. Furthermore, on the basis of the integral LNSG, a shear-locking-free finite-element (FE) model of the LNSG Timoshenko beams is constructed by introducing a new efficient higher order beam element with simple shape functions which can consider the influence of strains gradient as well as maintain the shear-locking-free property. Agreement between the exact results obtained from the differential LNSG and those of the FE model, integral LNSG, reveals that the LNSG is consistent and can be utilized instead of the common nonlocal strain gradient elasticity theory.

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Authors and Affiliations

  1. School of Mechanical Engineering, Iran University of Science and Technology, Narmak, 16846-13114, Tehran, Iran

    Mahmood Fakher & Shahrokh Hosseini-Hashemi

  2. Center of Excellence in Railway Transportation, Iran University of Science and Technology, Narmak, 16842-13114, Tehran, Iran

    Shahrokh Hosseini-Hashemi

Authors
  1. Mahmood Fakher
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Correspondence to Mahmood Fakher.

Appendix-A

Appendix-A

Non dimensional form of CBCs related to the TBT:

$$\begin{gathered} \frac{1}{{\Gamma_{1}^{2} \Gamma_{2} }}(( - 1 + \zeta_{1} )\Gamma_{2} \Gamma_{4}^{2} \partial_{x} \Phi - ( - 1 + \zeta_{1} )\Gamma_{1} \Gamma_{2} \Gamma_{4}^{2} \partial_{x,x} \Phi - \hfill \\ ( - 1 + \zeta_{1} )\Gamma_{1}^{2} \Gamma_{2} (\partial_{x} \Phi - \Gamma_{4}^{2} \partial_{x,x,x} \Phi ) + ( - 1 + \zeta_{1} )\Gamma_{1}^{3} \Gamma_{2} (\partial_{x,x} \Phi - \Gamma_{4}^{2} \partial_{x,x,x,x} \Phi ) - \hfill \\ \Gamma_{1}^{4} (\lambda^{2} \partial_{x} \Phi + \Gamma_{2} (\lambda^{2} \overline{W}_{T} + \zeta_{1} \partial_{x,x,x} \Phi - \zeta_{1} \Gamma_{4}^{2} \partial_{x,x,x,x,x} \Phi )) + \hfill \\ \Gamma_{1}^{5} (\lambda^{2} \partial_{x,x} \Phi + \Gamma_{2} (\lambda^{2} \partial_{x} \overline{W}_{T} + \zeta_{1} \partial_{x,x,x,x} \Phi - \zeta_{1} \Gamma_{4}^{2} \partial_{x,x,x,x,x,x} \Phi )) + \hfill \\ (\Gamma_{2} \Gamma_{4}^{2} - \zeta_{1} \Gamma_{2} \Gamma_{4}^{2} )\partial_{x} \Phi ) = 0\,\,\,at\,\overline{x} = 0 \hfill \\ \end{gathered}$$
(64)
$$\begin{gathered} \frac{1}{{\Gamma_{1}^{2} \Gamma_{2} }}( - ( - 1 + \zeta_{1} )\Gamma_{2} \Gamma_{4}^{2} \partial_{x} \Phi - ( - 1 + \zeta_{1} )\Gamma_{1} \Gamma_{2} \Gamma_{4}^{2} \partial_{x,x} \Phi + \hfill \\ ( - 1 + \zeta_{1} )\Gamma_{1}^{2} \Gamma_{2} (\partial_{x} \Phi - \Gamma_{4}^{2} \partial_{x,x,x} \Phi ) + ( - 1 + \zeta_{1} )\Gamma_{1}^{3} \Gamma_{2} (\partial_{x,x} \Phi - \Gamma_{4}^{2} \partial_{x,x,x,x} \Phi ) + \hfill \\ \Gamma_{1}^{4} (\lambda^{2} \partial_{x} \Phi + \Gamma_{2} (\lambda^{2} \overline{W}_{T} + \zeta_{1} \partial_{x,x,x} \Phi - \zeta_{1} \Gamma_{4}^{2} \partial_{x,x,x,x,x} \Phi )) + \hfill \\ \Gamma_{1}^{5} (\lambda^{2} \partial_{x,x} \Phi + \Gamma_{2} (\lambda^{2} \partial_{x} \overline{W}_{T} + \zeta_{1} \partial_{x,x,x,x} \Phi - \zeta_{1} \Gamma_{4}^{2} \partial_{x,x,x,x,x,x} \Phi )) + \hfill \\ ( - \Gamma_{2} \Gamma_{4}^{2} + \zeta_{1} \Gamma_{2} \Gamma_{4}^{2} )\partial_{x} \Phi ) = 0\,\,at\,\overline{x} = 1 \hfill \\ \end{gathered}$$
(65)
$$\begin{gathered} \frac{1}{{\Gamma_{1}^{3} }}(( - 1 + \zeta_{1} )\Gamma_{4}^{2} (\Phi + \partial_{x} \overline{W}_{T} ) - ( - 1 + \zeta_{1} )\Gamma_{1} \Gamma_{4}^{2} (\partial_{x} \Phi + \partial_{x,x} \overline{W}_{T} ) - \hfill \\ ( - 1 + \zeta_{1} )\Gamma_{1}^{2} (\Phi + \partial_{x} \overline{W}_{T} - \Gamma_{4}^{2} (\partial_{x,x} \Phi + \partial_{x,x,x} \overline{W}_{T} )) + \hfill \\ ( - 1 + \zeta_{1} )\Gamma_{1}^{3} (\partial_{x} \Phi + \partial_{x,x} \overline{W}_{T} - \Gamma_{4}^{2} (\partial_{x,x,x} \Phi + \partial_{x,x,x,x} \overline{W}_{T} )) - \hfill \\ \Gamma_{1}^{4} (\lambda^{2} \Gamma_{3} \partial_{x} \overline{W}_{T} + \zeta_{1} (\partial_{x,x} \Phi + \partial_{x,x,x} \overline{W}_{T} - \Gamma_{4}^{2} (\partial_{x,x,x,x} \Phi + \partial_{x,x,x,x,x} \overline{W}_{T} ))) + \hfill \\ \Gamma_{1}^{5} (\lambda^{2} \Gamma_{3} \partial_{x,x} \overline{W}_{T} + \zeta_{1} (\partial_{x,x,x} \Phi + \partial_{x,x,x,x} \overline{W}_{T} - \Gamma_{4}^{2} (\partial_{x,x,x,x,x} \Phi + \partial_{x,x,x,x,x,x} \overline{W}_{T} ))) + \hfill \\ (\Gamma_{4}^{2} - \zeta_{1} \Gamma_{4}^{2} )(\Phi + \partial_{x} \overline{W}_{T} ))\, = 0\,\,\,at\,\overline{x} = 0 \hfill \\ \end{gathered}$$
(66)
$$\begin{gathered} \frac{1}{{\Gamma_{1}^{3} }}( - ( - 1 + \zeta_{1} )\Gamma_{4}^{2} (\Phi + \partial_{x} \overline{W}_{T} ) - ( - 1 + \zeta_{1} )\Gamma_{1} \Gamma_{4}^{2} (\partial_{x} \Phi + \partial_{x,x} \overline{W}_{T} ) + \hfill \\ ( - 1 + \zeta_{1} )\Gamma_{1}^{2} (\Phi + \partial_{x} \overline{W}_{T} - \Gamma_{4}^{2} (\partial_{x,x} \Phi + \partial_{x,x,x} \overline{W}_{T} )) + \hfill \\ ( - 1 + \zeta_{1} )\Gamma_{1}^{3} (\partial_{x} \Phi + \partial_{x,x} \overline{W}_{T} - \Gamma_{4}^{2} (\partial_{x,x,x} \Phi + \partial_{x,x,x,x} \overline{W}_{T} )) + \hfill \\ \Gamma_{1}^{4} (\lambda^{2} \Gamma_{3} \partial_{x} \overline{W}_{T} + \zeta_{1} (\partial_{x,x} \Phi + \partial_{x,x,x} \overline{W}_{T} - \Gamma_{4}^{2} (\partial_{x,x,x,x} \Phi + \partial_{x,x,x,x,x} \overline{W}_{T} ))) + \hfill \\ \Gamma_{1}^{5} (\lambda^{2} \Gamma_{3} \partial_{x,x} \overline{W}_{T} + \zeta_{1} (\partial_{x,x,x} \Phi + \partial_{x,x,x,x} \overline{W}_{T} - \Gamma_{4}^{2} (\partial_{x,x,x,x,x} \Phi + \partial_{x,x,x,x,x,x} \overline{W}_{T} ))) + \hfill \\ ( - \Gamma_{4}^{2} + \zeta_{1} \Gamma_{4}^{2} )(\Phi + \partial_{x} \overline{W}_{T} )) = 0\,\,\,at\,\overline{x} = 1. \hfill \\ \end{gathered}$$
(67)

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Fakher, M., Hosseini-Hashemi, S. On the vibration of nanobeams with consistent two-phase nonlocal strain gradient theory: exact solution and integral nonlocal finite-element model. Engineering with Computers 38, 2361–2384 (2022). https://doi.org/10.1007/s00366-020-01206-5

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