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Small scale effects on transient vibrations of porous FG cylindrical nanoshells based on nonlocal strain gradient theory

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Abstract

This research investigates transient vibrational characteristics of a porous functionally graded cylindrical nanoshell under different impulsive loadings with the use of nonlocal strain gradient theory (NSGT). Based on NSGT, two size parameters accounting for stiffness softening and hardening effects are incorporated in modeling of the nanoshell. Impulse forces have three forms of triangular, rectangular and sinusoidal. Two sorts of porosity distributions called even and uneven have been taken into account. Governing equations obtained for porous nanoshell have been solved through inverse Laplace transforms technique to derive dynamical deflections. It is shown that transient responses of a nanoshell are affected by the form and position of impulse loading, amount of porosities, porosities dispensation, nonlocal and strain gradient parameters.

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Author information

Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Qatar University, P.O. Box 2713, Doha, Qatar

    Masoud Forsat, Seyed Sajad Mirjavadi & A. M. S. Hamouda

  2. Department of Electrical Engineering, Qatar University, Doha, Qatar

    Salman Badnava

  3. Aerospace Engineering Department, Center of Excellence in Computational Aerospace, AmirKabir University of Technology, Tehran, 15875-4413, Iran

    Mohammad Reza Barati

Authors
  1. Masoud Forsat
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  2. Salman Badnava
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  3. Seyed Sajad Mirjavadi
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  4. Mohammad Reza Barati
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  5. A. M. S. Hamouda
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Corresponding author

Correspondence to Seyed Sajad Mirjavadi.

Appendix

Appendix

$$ k_{1,1} = A_{11} \left( {\varPsi_{31} - l^{2} \left( {\varPsi_{51} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{31} } \right)} \right) - n^{2} \frac{{A_{66} }}{{R^{2} }}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), $$
(44)
$$\begin{aligned} & k_{2,1} = n\left( {\frac{{A_{12} }}{R} + \frac{{A_{66} }}{R}} \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\\ & \qquad \;k_{1,2} = - n\left( {\frac{{A_{12} }}{R} + \frac{{A_{66} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\end{aligned} $$
(45)
$$ k_{3,1} = + \frac{{A_{12} }}{R}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), \;k_{1,3} = - \frac{{A_{12} }}{R}\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right), $$
(46)
$$ \begin{aligned} k_{4,1} = & + B_{11} \left( {\varPsi_{31} - l^{2} \left( {\varPsi_{51} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{31} } \right)} \right) - n^{2} \frac{{B_{66} }}{{R^{2} }}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), \\ k_{1,4} = & B_{11} \left( {\varPsi_{31} - l^{2} \left( {\varPsi_{51} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{31} } \right)} \right) - n^{2} \frac{{B_{66} }}{{R^{2} }}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), \\ \end{aligned} $$
(47)
$$ \begin{aligned} & k_{5,1} = n\left( {\frac{{B_{12} }}{R} + \frac{{B_{66} }}{R}} \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\\ & \qquad \;k_{1,5} = - n\left( {\frac{{B_{66} }}{R} + \frac{{B_{12} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\end{aligned} $$
(48)
$$\begin{aligned} & k_{2,2} = A_{66} \left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right) - n^{2} \frac{{A_{11} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right) \\ & \qquad - \frac{{\tilde{A}_{66} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned} $$
(49)
$$\begin{aligned} & k_{3,2} = - n\left( {\frac{{A_{11} }}{{R^{2} }} + \frac{{\tilde{A}_{66} }}{{R^{2} }}} \right)\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\\ & \qquad \;k_{2,3} = - n\left( {\frac{{\tilde{A}_{66} }}{{R^{2} }} + \frac{{A_{11} }}{{R^{2} }}} \right)\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned} $$
(50)
$$\begin{aligned} & k_{4,2} = - n\left( {\frac{{B_{12} }}{R} + \frac{{B_{66} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\\ & \qquad \;k_{2,4} = + n\left( {\frac{{B_{12} }}{R} + \frac{{B_{66} }}{R}} \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\end{aligned} $$
(51)
$$\begin{aligned} k_{5,2} &= B_{66} \left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right) - n^{2} \frac{{B_{11} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right)\\ & \quad + \frac{{\tilde{A}_{66} }}{R}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right),} \right)\end{aligned} $$
(52)
$$\begin{aligned} k_{3,3} &= \tilde{A}_{66} \left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}}}}\varPsi_{20} } \right)} \right) - n^{2} \frac{{\tilde{A}_{66} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right) \\ & \quad- \frac{{A_{11} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned}$$
(53)
$$\begin{aligned} & k_{4,3} = \left( {\tilde{A}_{66} - \frac{{B_{12} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\\ & \qquad\;k_{3,4} = + \left( {\frac{{B_{12} }}{R} - \tilde{A}_{66} } \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\end{aligned} $$
(54)
$$\begin{aligned} & k_{5,3} = n\left( { + \frac{{\tilde{A}_{66} }}{R} - \frac{{B_{11} }}{{R^{2} }}} \right)\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\\ & \qquad \;k_{3,5} = - n\left( { + \frac{{B_{11} }}{{R^{2} }} - \frac{{\tilde{A}_{66} }}{R}} \right)\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned} $$
(55)
$$\begin{aligned} k_{4,4} & = + D_{11} \left( {\varPsi_{31} - l^{2} \left( {\varPsi_{51} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{31} } \right)} \right) - n^{2} \frac{{D_{66} }}{{R^{2} }}\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right)\\ &\quad - \tilde{A}_{66} \left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\end{aligned} $$
(56)
$$\begin{aligned} & k_{5,4} = + n\left( {\frac{{D_{12} }}{R} + \frac{{D_{66} }}{R}} \right)\left( {\varPsi_{11} - l^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\\ & \qquad\;k_{4,5} = - n\left( {\frac{{D_{66} }}{R} + \frac{{D_{12} }}{R}} \right)\left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right),\end{aligned} $$
(57)
$$\begin{aligned} k_{5,5}& = + D_{66} \left( {\varPsi_{20} - l^{2} \left( {\varPsi_{40} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{20} } \right)} \right) - n^{2} \frac{{D_{11} }}{{R^{2} }}\left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right)\\ & \quad - \tilde{A}_{66} \left( {\varPsi_{00} - l^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right),\end{aligned} $$
(58)
$$ m_{1,1} = + I_{0} \left( {\varPsi_{11} - ea^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), $$
(59)
$$ m_{2,2} = m_{3,3} = m_{5,5} = + I_{0} \left( {\varPsi_{00} - ea^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right), $$
(60)
$$ m_{4,4} = + I_{2} \left( {\varPsi_{11} - ea^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right), $$
(61)
$$ m_{4,1} = + I_{1} \left( {\varPsi_{11} - ea^{2} \left( {\varPsi_{31} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{11} } \right)} \right),\;m_{5,2} = m_{2,5} = + I_{1} \left( {\varPsi_{00} - ea^{2} \left( {\varPsi_{20} - \frac{{n^{2} }}{{R^{{}} }}\varPsi_{00} } \right)} \right), $$
(62)
$$ Q_{\text{dynamic}} = Q_{n} \varPsi_{00} , $$
(63)

in which

$$ \varPsi_{00} = \int_{0}^{L} {F_{m} } F_{m} {\text{d}}x, $$
(64)
$$ \varPsi_{20} = \int_{0}^{L} {\frac{{{\text{d}}^{2} F_{m}^{{}} }}{{{\text{d}}x^{2} }}} F_{m} {\text{d}}x, $$
(65)
$$ \varPsi_{11} = \int_{0}^{L} {\frac{{{\text{d}}F_{m}^{{}} }}{{{\text{d}}x}}} \frac{{{\text{d}}F_{m}^{{}} }}{{{\text{d}}x}}{\text{d}}x, $$
(66)
$$ \varPsi_{31} = \int\limits_{0}^{L} {\frac{{{\text{d}}^{3} F_{m}^{{}} }}{{{\text{d}}x^{3} }}} \frac{{{\text{d}}F_{m}^{{}} }}{{{\text{d}}x}}{\text{d}}x, $$
(67)
$$ \varPsi_{40} = \int_{0}^{L} {\frac{{{\text{d}}^{4} F_{m}^{{}} }}{{{\text{d}}x^{4} }}} F_{m} {\text{d}}x, $$
(68)
$$ \varPsi_{51} = \int_{0}^{L} {\frac{{{\text{d}}^{5} F_{m}^{{}} }}{{{\text{d}}x^{5} }}} \frac{{{\text{d}}F_{m}^{{}} }}{{{\text{d}}x}}{\text{d}}x, $$
(69)

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Forsat, M., Badnava, S., Mirjavadi, S.S. et al. Small scale effects on transient vibrations of porous FG cylindrical nanoshells based on nonlocal strain gradient theory. Eur. Phys. J. Plus 135, 81 (2020). https://doi.org/10.1140/epjp/s13360-019-00042-x

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