Nonlinear resonant behaviors of embedded thick FG double layered nanoplates via nonlocal strain gradient theory

Abstract

This research deals with the nonlinear forced vibration behavior of embedded functionally graded double layered nanoplates with all edges simply supported via nonlocal strain gradient elasticity theory based on the Mindlin plate theory along with von Kármán geometric nonlinearity. The interaction of van der Waals forces between adjacent layers is included. For modeling surrounding elastic medium, the nonlinear Winkler–Pasternak foundation model is employed. The exact solution of the nonlinear forced vibration for primary and secondary resonance through the Harmonic Balance method is then established. For the double layered nanoplate, uniform distribution and sinusoidal distribution of loading are considered. It is assumed that the material properties of functionally graded double layered nanoplates are graded in the thickness direction and estimated through the rule of mixture. The Galerkin-based numerical technique is employed to reduce the set of nonlinear governing equations into a time-varying set of ordinary differential equations. The effects of different parameters such as length scale parameters, elastic foundation parameters, dimensionless transverse force and gradient index on the frequency responses of functionally graded double layered nanoplates are investigated. The results show that the length scale parameters give nonlinearity of the hardening type in nonlinear forced vibration and the increase in strain gradient parameter causes to increase in maximum response amplitude, whereas the increase in nonlocal parameter causes to decrease in maximum response amplitude.

Appendix

$$ N_{i}^{*} = N_{i} - \frac{{D_{1} }}{{\kappa^{2} \nu_{1} B_{1} }}\nabla^{2} N_{i} + \frac{{m_{2} }}{{\kappa^{2} \nu_{1} B_{1} }}\left( {1 - \mu^{2} \nabla^{2} } \right)\ddot{N}_{i} ,\quad i = 1,2, $$

$$ f_{i}^{*} = q_{i} - \frac{{D_{1} }}{{\kappa^{2} \nu_{1} B_{1} }}\nabla^{2} q_{i} + \frac{{m_{2} }}{{\kappa^{2} \nu_{1} B_{1} }}\left( {1 - \mu^{2} \nabla^{2} } \right)\ddot{q}_{i} ,\quad i = 1,2, $$

$$ C_{i}^{*} = C_{i} - \frac{{cD_{1} }}{{\kappa^{2} \nu_{1} B_{1} }}\nabla^{2} C_{i} + \frac{{m_{2} }}{{\kappa^{2} \nu_{1} B_{1} }}\left( {1 - \mu^{2} \nabla^{2} } \right)\ddot{C}_{i} ,\quad i = 1,2, $$

$$ N_{i} = \frac{{\partial^{2} \phi_{i} }}{{\partial y^{2} }}\frac{{\partial^{2} w_{i} }}{{\partial x^{2} }} + \frac{{\partial^{2} \phi_{i} }}{{\partial x^{2} }}\frac{{\partial^{2} w_{i} }}{{\partial y^{2} }} - 2\frac{{\partial^{2} \phi_{i} }}{\partial x\partial y}\frac{{\partial^{2} w_{i} }}{\partial x\partial y},\quad i = 1,2, $$

$$ C_{1} = c\left( {w_{1} - w_{2} } \right),\,C_{2} = c\left( {w_{2} - w_{1} } \right), $$

$$ q_{1} = F\left( {x,y,t} \right) - k_{l} w_{1} - k_{nl} w_{1}^{3} + k_{s} \nabla^{2} w_{1} , $$

$$ q_{2} = - k_{l} w_{2} - k_{nl} w_{2}^{3} + k_{s} \nabla^{2} w_{2} , $$

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$$ \alpha_{1} = \frac{{a^{2} b^{2} \left( {m_{0} + m_{2} (c + k_{l} )\hat{G}} \right) + \left( {a^{2} + b^{2} } \right)\pi^{2} \left( {m_{0} (\hat{G}D_{1} + \mu^{2} ) + k_{s} m_{2} \hat{G} + m_{2} + m_{2} \mu^{2} (c + k_{l} + \hat{A}k_{s} )\hat{G}} \right)}}{{m_{0} m_{2} \hat{G}\left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right]}}, $$

$$ \alpha_{2} = \frac{{3\left[ {A_{1} \left( {a^{4} + b^{4} } \right)\pi^{4} + 9a^{4} b^{4} k_{NL} } \right]}}{{16a^{4} b^{4} m_{0} }}, $$

$$ \alpha_{3} = \frac{{a^{4} b^{4} \left( {c + k_{l} } \right) + \hat{A}a^{4} b^{4} \left[ {k_{l} \hat{G}D_{1} + k_{s} + k_{l} \mu^{2} + c\left( {\hat{G}D_{1} + \mu^{2} } \right)} \right]}}{{m_{0} m_{2} \hat{G}\left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right]^{2} }} + \frac{{\hat{A}^{2} a^{4} b^{4} \left[ {k_{s} \mu^{2} + D_{1} \left( {1 + \hat{G}\left( {k_{s} + \left( {c + k_{l} } \right)\mu^{2} } \right)} \right) + \hat{A}D_{1} (\hat{G}k_{s} \mu^{2} + l^{2} )} \right]}}{{m_{0} m_{2} \hat{G}\left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right]^{2} }}, $$

$$ \alpha_{4} = \frac{{6\left[ {A_{1} \left( {a^{4} + b^{4} } \right)\pi^{4} + 9a^{4} b^{4} k_{NL} } \right]}}{{16a^{4} b^{4} m_{0} }}, $$

$$ \alpha_{5} = \frac{{\left[ {A_{1} \left( {a^{4} + b^{4} } \right)\pi^{4} + 9a^{4} b^{4} k_{NL} } \right]\left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} (\hat{G}D_{1} + \mu^{2} )} \right]}}{{16a^{4} b^{4} m_{0} m_{2} \hat{G}\left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right]}}, $$

$$ \alpha_{6} = - \frac{c}{{m_{0} }}, $$

$$ \alpha_{7} = - \frac{{c\left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} (\hat{G}D_{1} + \mu^{2} )} \right]}}{{m_{0} m_{2} \hat{G}\left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right]}}, $$

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$$ \alpha_{1}^{*} = 16a^{4} b^{4} m_{0} \left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right], $$

$$ \alpha_{2}^{*} = 16a^{2} b^{2} \left[ {a^{4} b^{4} c + \hat{A}^{3} \pi^{6} a^{4} b^{4} D_{1} l^{2} + ca^{2} b^{2} \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right], $$

$$ \alpha_{3}^{*} = A_{1} \pi^{2} \left( {a^{4} + b^{4} } \right)\left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right], $$

$$ \alpha_{4}^{*} = - 16a^{4} b^{4} c\left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right], $$

$$ \alpha_{5}^{*} = 16a^{6} b^{6} \left[ {a^{2} b^{2} + \left( {a^{2} + b^{2} } \right)\pi^{2} \mu^{2} } \right]. $$

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