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Wave dispersion characteristics of axially loaded magneto-electro-elastic nanobeams

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Abstract

The analysis of wave propagation behavior of a magneto-electro-elastic functionally graded (MEE-FG) nanobeam is performed in the framework of classical beam theory. To capture small-scale effects, the nonlocal elasticity theory of Eringen is applied. Furthermore, the material properties of nanobeam are assumed to vary gradually through the thickness based on power-law form. Nonlocal governing equations of MEE-FG nanobeam have been derived employing Hamilton’s principle. The results of present research have been validated by comparing with those of previous investigations. An analytical solution of governing equations is utilized to obtain wave frequencies, phase velocities and escape frequencies. Effects of various parameters such as wave number, nonlocal parameter, gradient index, axial load, magnetic potential and electric voltage on wave dispersion characteristics of MEE-FG nanoscale beams are studied in detail.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International University, Qazvin, Iran

    Farzad Ebrahimi, Mohammad Reza Barati & Ali Dabbagh

Authors
  1. Farzad Ebrahimi
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  2. Mohammad Reza Barati
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  3. Ali Dabbagh
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Correspondence to Farzad Ebrahimi.

Appendix

Appendix

In Eq. (55), \(a_{ij}\) and \(m_{ij} , \left( {i,j = 1,2,3,4} \right)\) are defined as follows:

$$\begin{aligned} a_{11} = - A_{11} k^{2} \hfill \\ a_{12} = - iB_{11} k^{3} \hfill \\ a_{13} = ikA_{31}^{e} \hfill \\ a_{14} = ikA_{31}^{m} \hfill \\ a_{21} = iB_{11} k^{3} \hfill \\ a_{22} = - D_{11} k^{4} + \left( {1 + \mu k^{2} } \right)\left[ {\left( {\tilde{N} + N^{E} + N^{H} } \right)k^{2} } \right] \hfill \\ a_{23} = + E_{31}^{e} k^{2} \hfill \\ a_{24} = + E_{31}^{m} k^{2} \hfill \\ a_{31} = ikA_{31}^{e} \hfill \\ a_{32} = - E_{31}^{e} k^{2} \hfill \\ a_{33} = - \left( {F_{33}^{e} + F_{11}^{e} k^{2} } \right) \hfill \\ a_{34} = - \left( {F_{31}^{m} + F_{11}^{m} k^{2} } \right) \hfill \\ a_{41} = ikA_{31}^{m} \hfill \\ a_{42} = E_{31}^{m} k^{2} \hfill \\ a_{43} = - \left( {F_{31}^{m} + F_{11}^{m} k^{2} } \right) \hfill \\ a_{44} = - \left( {X_{33}^{m} + X_{11}^{m} k^{2} } \right) \hfill \\ \end{aligned}$$
(59)

and

$$\begin{aligned} m_{11} = I_{0} \left( {1 + \mu k^{2} } \right) \hfill \\ m_{12} = I_{1} \left( {1 + \mu k^{2} } \right) \hfill \\ m_{13} = m_{14} = 0 \hfill \\ m_{21} = - iI_{1} k\left( {1 + \mu k^{2} } \right) \hfill \\ m_{22} = + \left( {I_{0} + I_{2} k^{2} } \right)\left( {1 + \mu k^{2} } \right) \hfill \\ m_{23} = m_{24} = m_{31} = m_{32} = m_{33} = m_{34} = 0 \hfill \\ \end{aligned}$$
(60)

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Ebrahimi, F., Barati, M.R. & Dabbagh, A. Wave dispersion characteristics of axially loaded magneto-electro-elastic nanobeams. Appl. Phys. A 122, 949 (2016). https://doi.org/10.1007/s00339-016-0465-1

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