Energy absorption, free and forced vibrations of flexoelectric nanocomposite magnetostrictive sandwich nanoplates with single sinusoidal edge on the frictional torsional viscoelastic medium

Abstract

Most studies on the nanoscale mainly focus on regular rectangular nanoplates, but according to the synthesis of nanostructures, the dynamic response of non-rectangular nanoplates is noticeable and there are not many works on these complex nanostructures. This work presents energy absorption, and forced and free vibrations of sandwich non-rectangular nanoplates with a single sinusoidal edge resting on a fractional torsional viscoelastic medium. The nanostructure is made from alumina reinforced by graphene platelets (GPLs) as a core covered by the flexoelectric and magnetostrictive materials as top and bottom layers, respectively. The consideration of size effects is derived from the innovative theory of local/nonlocal phenomena in a two-phase context. The Halpin–Tsai micromechanical and Kelvin–Voigt models are applied for the effective characteristics of the material in the nanocomposite layer and structural damping, respectively. Based on Hamilton’s principle and refined zigzag theory (RZT), the coupled electro-magneto-mechanical equations of motion are gained and analyzed by Galerkin’s and Newmark’s procedures. The effects of different components, including factors related to both the nonlocal and local phase fractions, the volume fraction of GPLs, various elastic mediums, electric field, structural damping, magnetic field, piezoelectric and flexoelectric effects on the absorption of energy, and forced and free vibrations of the sandwich nanostructure. Numerical simulations demonstrate that optimal energy absorption occurs when the flexoelectric factor is set to zero and the piezoelectric constant is non-zero but of opposite polarity. Additionally, it is concluded that when the coefficient of the local phase fraction is zero, increasing the nonlocal factor has more influence on the energy absorption and vibration of the nanostructure.

Appendices

Appendix 1

$$ m_{11} = - \frac{{d_{31} }}{{a_{33} }}, $$

(96)

$$ m_{12} = - \frac{{d_{31} }}{{a_{33} }}, $$

(97)

$$ \begin{gathered} m_{13} = - \frac{1}{{c_{11}^{{\text{t}}} \left( {a_{33} k + 1} \right)c_{11}^{{\text{b}}} c_{11}^{{\text{c}}} a_{33} }}\left( {\left( {\frac{{d_{31} c_{11}^{{\text{c}}} \left( {G_{x} - c_{11}^{{\text{t}}} } \right)h_{{\text{t}}} }}{2} + } \right.} \right.\left( {\left( {\left( { - k\left( {z + h} \right)a_{33} - h - \frac{{h_{{\text{c}}} }}{2}} \right)d_{31} - a_{33} f_{31} k - f_{31} } \right)} \right.c_{11}^{{\text{t}}} \hfill \\ + \left. {\left. {\left( {\left( {k\left( {h - 2h_{{\text{c}}} + z} \right)a_{33} + h - 2h_{{\text{c}}} + \frac{{h_{{\text{c}}} }}{2}} \right)d_{31} + f_{31} \left( {a_{33} k + 1} \right)} \right)G_{x} } \right)c_{11}^{{\text{c}}} + 2d_{31} G_{x} c_{11}^{t} \left( {h_{{\text{c}}} - h_{{\text{b}}} } \right)\left( {a_{33} k + 1} \right)} \right)c_{11}^{{\text{b}}} \hfill \\ + \left. {2d_{31} G_{x} h_{{\text{b}}} c_{11}^{{\text{t}}} c_{11}^{{\text{c}}} \left( {a_{33} k + 1} \right)} \right) \hfill \\ \end{gathered} $$

(98)

$$ \begin{gathered} m_{14} = - \frac{1}{{c_{{_{22} }}^{{\text{t}}} \left( {a_{33} k + 1} \right)c_{22}^{{\text{b}}} c_{22}^{{\text{c}}} a_{33} }}\left( {\left( {\frac{{d_{31} c_{22}^{{\text{c}}} \left( {G_{x} - c_{22}^{{\text{t}}} } \right)h_{t} }}{2} + } \right.} \right.\left( {\left( {\left( { - k\left( {z + h} \right)a_{33} - h - \frac{{h_{{\text{c}}} }}{2}} \right)d_{31} - a_{33} f_{31} k - f_{31} } \right)} \right.c_{22}^{{\text{t}}} \hfill \\ + \left. {\left. {\left( {k\left( {\left( {z + h} \right)G_{x} - 2G_{y} h_{{\text{c}}} } \right)a_{33} + \left( {h + \frac{{h_{{\text{c}}} }}{2}} \right)G_{x} - 2G_{y} h_{{\text{c}}} } \right)d_{31} + f_{31} G_{x} \left( {a_{33} k + 1} \right)} \right)c_{22}^{{\text{c}}} + 2d_{31} G_{x} c_{22}^{{\text{t}}} \left( {h_{{\text{c}}} - h_{{\text{b}}} } \right)\left( {a_{33} k + 1} \right)} \right)c_{22}^{{\text{b}}} \hfill \\ + \left. {2d_{31} G_{y} h_{{\text{b}}} c_{22}^{{\text{t}}} c_{22}^{{\text{c}}} \left( {a_{33} k + 1} \right)} \right) \hfill \\ \end{gathered} $$

(99)

$$ m_{15} = - \frac{{\left( {\frac{{h{}_{{\text{t}}}d_{31} }}{2} + \left( {\frac{{h{}_{{\text{c}}}}}{2} + a_{33} k\,z} \right)d_{31} + f_{31} \left( {a_{33} k + 1} \right)} \right)}}{{\left( {a_{33} k + 1} \right)a_{33} }}, $$

(100)

$$ m_{16} = - \frac{{\left( {\frac{{h_{{\text{t}}}d_{31} }}{2} + \left( {\frac{{h_{{\text{c}}}}}{2} + a_{33} k\,z} \right)d_{31} + f_{31} \left( {a_{33} k + 1} \right)} \right)}}{{\left( {a_{33} k + 1} \right)a_{33} }}, $$

(101)

$$ m_{17} = - \frac{{v_{0} }}{{h_{{\text{t}}} a_{33} }}, $$

(102)

Appendix 2

$$ \begin{gathered} L_{11} = \left( {\frac{{A_{11}^{c} m^{2} \pi^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)b}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{2} }} + \zeta k^{2} \left( {\frac{{D_{11x} m^{4} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)b}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} }} + \frac{{A_{11}^{b} m^{4} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)b}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} }}} \right.} \right. \hfill \\ + \frac{{A_{11}^{c} m^{4} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)b}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} }} + D_{11x} \left( { - \frac{{3m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} }}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b}}} \right. - \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)fc^{2} \sin \left( {\frac{c\pi y}{b}} \right)}}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{3} b}} \hfill \\ + \left. {\frac{{m^{4} \pi^{6} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{6} b}} + \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{2} }}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{2} b}}} \right) + A_{11}^{b} \left( { - \frac{{3m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} }}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b}}} \right. \hfill \\ - \left. {\frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)fc^{2} \sin \left( {\frac{c\pi y}{b}} \right)}}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{3} b}} + \frac{{m^{4} \pi^{6} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{6} b}} + \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{2} }}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{2} b}}} \right) \hfill \\ + A_{11}^{c} \left( { - \frac{{3m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} }}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b}}} \right. - \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)fc^{2} \sin \left( {\frac{c\pi y}{b}} \right)}}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{3} b}} \hfill \\ + \frac{{m^{4} \pi^{6} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{6} b}} + \left. {\left. {\frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{2} }}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{2} b}}} \right)} \right) \hfill \\ + \zeta k^{2} \left( {A_{66}^{b} \left( { - \frac{{18m^{2} \pi^{6} x^{2} f^{4} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{6} b^{3} }}} \right.} \right. + \frac{{m^{4} \pi^{8} x^{4} f^{4} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{8} b^{3} }} \hfill \\ \end{gathered} $$

$$ \begin{gathered} \frac{{3m^{2} \pi^{6} x^{2} f^{2} c^{4} \sin \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b^{3} }} - \frac{{18m^{2} \pi^{6} x^{2} f^{3} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)\sin \left( {\frac{c\pi y}{b}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{5} b^{3} }} + \frac{{2m^{2} \pi^{6} x^{2} f^{2} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b^{3} }} \hfill \\ + \left. {\frac{{3m^{2} \pi^{6} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{2} }}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b^{3} }} + \frac{{\left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{4} \pi^{4} }}{{2b^{3} }}} \right) + D_{11xy} \left( { - \frac{{18m^{2} \pi^{6} x^{2} f^{4} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{6} b^{3} }}} \right. + \hfill \\ \frac{{m^{4} \pi^{8} x^{4} f^{4} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{8} b^{3} }} - \frac{{3m^{2} \pi^{6} x^{2} f^{2} c^{4} \sin \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b^{3} }} - \hfill \\ \end{gathered} $$

$$ \begin{gathered} \frac{{18m^{2} \pi^{6} x^{2} f^{3} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)\sin \left( {\frac{c\pi y}{b}} \right)}}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{5} b^{3} }} + \frac{{2m^{2} \pi^{6} x^{2} f^{2} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b^{3} }} + \hfill \\ \left. {\frac{{3m^{2} \pi^{6} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{2} }}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b^{3} }} + \frac{{\left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{4} \pi^{4} }}{{2b^{3} }}} \right) + D_{11xy} \hfill \\ \left( { - \frac{{3m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)f^{2} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{2} }}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b}}} \right. - \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)f^{{}} c^{2} \sin \left( {\frac{c\pi y}{b}} \right)}}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{3} b}} + \hfill \\ \left. {\frac{{m^{4} \pi^{6} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{6} b}} + \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{2} }}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{2} b}}} \right) + A_{66}^{c} \left( { - \frac{{3m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)f^{2} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{2} }}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b}}} \right. \hfill \\ \left. { - \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)f^{{}} c^{2} \sin \left( {\frac{c\pi y}{b}} \right)}}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{3} b}} + \frac{{m^{4} \pi^{6} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{6} b}} + \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{2} }}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{2} b}}} \right) + A_{11}^{b} \hfill \\ \hfill \\ \end{gathered} $$

$$ \begin{gathered} \left( { - \frac{{3m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)f^{2} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{2} }}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b}}} \right. - \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)f^{{}} c^{2} \sin \left( {\frac{c\pi y}{b}} \right)}}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{3} b}} + \frac{{m^{4} \pi^{6} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{6} b}} \hfill \\ \left. { + \frac{{m^{2} \pi^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{2} }}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{2} b}}} \right) + A_{66}^{c} \left( { - \frac{{18m^{2} \pi^{6} x^{2} f^{4} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{6} b^{3} }} + \frac{{m^{4} \pi^{8} x^{4} f^{4} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{4} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{8} b^{3} }}} \right. \hfill \\ - \frac{{3m^{2} \pi^{6} x^{2} f^{2} c^{4} \sin \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b^{3} }} - \frac{{18m^{2} \pi^{6} x^{2} f^{3} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)\sin \left( {\frac{c\pi y}{b}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{5} b^{3} }} \hfill \\ + \left. {\left. {\frac{{2m^{2} \pi^{6} x^{2} f^{2} c^{4} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b^{3} }} + \frac{{3m^{2} \pi^{6} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{2} }}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b^{3} }} + \frac{{\left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{4} \pi^{4} }}{{2b^{3} }}} \right)} \right) \hfill \\ + \frac{{D_{15x} m^{2} \pi^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)b}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{2} }} + \frac{{Q_{11} m^{2} \pi^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)b}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{2} }} - A_{66}^{b} \left( { - \frac{{m^{2} \pi^{4} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b}}} \right. \hfill \\ - \left. {\frac{{\left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{4} \pi^{2} }}{2b}} \right) - A_{66}^{c} \left( { - \frac{{m^{2} \pi^{4} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b}} - \frac{{\left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{4} \pi^{2} }}{2b}} \right) - D_{11xy} \hfill \\ \left( { - \frac{{m^{2} \pi^{4} x^{2} f^{2} c^{2} \cos \left( {\frac{c\pi y}{b}} \right)^{2} \left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)}}{{2\left( {a + f\sin \left( {\frac{c\pi y}{b}} \right)} \right)^{4} b}} - \frac{{\left( {\frac{a}{2} + \frac{{f\sin \left( {\frac{c\pi y}{b}} \right)}}{2}} \right)n^{4} \pi^{2} }}{2b}} \right) \hfill \\ \end{gathered} $$

where

$$ A_{11}^{{\text{c}}} = \left( {1 + g\frac{\partial }{\partial t}} \right)\int\limits_{{ - h_{{\text{c}}} /2}}^{{h_{{\text{c}}} /2}} {c_{11}^{{\text{c}}} {\text{d}}z} $$

(103)

$$ A_{11}^{{\text{b}}} = \left( {1 + g\frac{\partial }{\partial t}} \right)\int\limits_{{ - h_{{\text{c}}} /2 - h_{{\text{b}}} }}^{{ - h_{{\text{c}}} /2}} {c_{11}^{{\text{b}}} {\text{d}}z} $$

(104)

$$ A_{66}^{{\text{c}}} = \left( {1 + g\frac{\partial }{\partial t}} \right)\int\limits_{{ - h_{{\text{c}}} /2}}^{{h_{{\text{c}}} /2}} {c_{66}^{{\text{c}}} {\text{d}}z} $$

(105)

$$ A_{66}^{{\text{b}}} = \left( {1 + g\frac{\partial }{\partial t}} \right)\int\limits_{{ - h_{{\text{c}}} /2 - h_{{\text{b}}} }}^{{ - h_{{\text{c}}} /2}} {c_{66}^{{\text{b}}} {\text{d}}z} $$

(106)

$$ D_{11x} = \int\limits_{{h_{c} /2}}^{{h_{{\text{c}}} /2 + h_{{\text{t}}} }} { - \left( { - \frac{{d_{31} d_{31} }}{{a_{33} }} + f_{13} \frac{\partial }{\partial z}\left( {\frac{{d_{31} }}{{a_{33} }}} \right) - c_{11}^{{\text{t}}} } \right){\text{d}}z} $$

(107)

$$ D_{11xy} = \left( {1 + g\frac{\partial }{\partial t}} \right)\int\limits_{{h_{{\text{c}}} /2}}^{{h_{{\text{c}}} /2 + h_{{\text{t}}} }} {c_{66}^{{\text{t}}} {\text{d}}z} $$

(108)

Appendix 3

$$ \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\left( {N_{11} \,u_{0} \left( t \right) + N_{12} \,v_{0} \left( t \right) + N_{13} \,w_{0} \left( t \right) + N_{14} \,\varphi_{0x} \left( t \right) + N_{15} \,\varphi_{0y} \left( t \right) + N_{16} \,\psi_{0x} \left( t \right) + N_{17} \,\psi_{0x} \left( t \right) + N_{18} \,\phi_{0} \left( t \right)} \right)} } = 0 $$

(109)

$$ \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\left( {N_{21} \,u_{0} \left( t \right) + N_{22} \,v_{0} \left( t \right) + N_{23} \,w_{0} \left( t \right) + N_{24} \,\varphi_{0x} \left( t \right) + N_{25} \,\varphi_{0y} \left( t \right) + N_{26} \,\psi_{0x} \left( t \right) + N_{27} \,\psi_{0x} \left( t \right) + N_{28} \,\phi_{0} \left( t \right)} \right)} } = 0 $$

(110)

$$ \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\left( {N_{31} \,u_{0} \left( t \right) + N_{32} \,v_{0} \left( t \right) + N_{33} \,w_{0} \left( t \right) + N_{34} \,\varphi_{0x} \left( t \right) + N_{35} \,\varphi_{0y} \left( t \right) + N_{36} \,\psi_{0x} \left( t \right) + N_{37} \,\psi_{0x} \left( t \right) + N_{38} \,\phi_{0} \left( t \right)} \right)} } = 0 $$

(111)

$$ \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\left( {N_{41} \,u_{0} \left( t \right) + N_{42} \,v_{0} \left( t \right) + N_{43} \,w_{0} \left( t \right) + N_{44} \,\varphi_{0x} \left( t \right) + N_{45} \,\varphi_{0y} \left( t \right) + N_{46} \,\psi_{0x} \left( t \right) + N_{47} \,\psi_{0x} \left( t \right) + N_{48} \,\phi_{0} \left( t \right)} \right)} } = 0 $$

(112)

$$ \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\left( {N_{51} \,u_{0} \left( t \right) + N_{52} \,v_{0} \left( t \right) + N_{53} \,w_{0} \left( t \right) + N_{54} \,\varphi_{0x} \left( t \right) + N_{55} \,\varphi_{0y} \left( t \right) + N_{56} \,\psi_{0x} \left( t \right) + N_{57} \,\psi_{0x} \left( t \right) + N_{58} \,\phi_{0} \left( t \right)} \right)} } = 0 $$

(113)

$$ \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\left( {N_{61} \,u_{0} \left( t \right) + N_{62} \,v_{0} \left( t \right) + N_{63} \,w_{0} \left( t \right) + N_{64} \,\varphi_{0x} \left( t \right) + N_{65} \,\varphi_{0y} \left( t \right) + N_{66} \,\psi_{0x} \left( t \right) + N_{67} \,\psi_{0x} \left( t \right) + N_{68} \,\phi_{0} \left( t \right)} \right)} } = 0 $$

(114)

$$ \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\left( {N_{71} \,u_{0} \left( t \right) + N_{72} \,v_{0} \left( t \right) + N_{73} \,w_{0} \left( t \right) + N_{74} \,\varphi_{0x} \left( t \right) + N_{75} \,\varphi_{0y} \left( t \right) + N_{76} \,\psi_{0x} \left( t \right) + N_{77} \,\psi_{0x} \left( t \right) + N_{78} \,\phi_{0} \left( t \right)} \right)} } = 0 $$

(C115)

$$ \sum\limits_{m = 1}^{M} {\sum\limits_{n = 1}^{N} {\left( {N_{81} \,u_{0} \left( t \right) + N_{82} \,v_{0} \left( t \right) + N_{83} \,w_{0} \left( t \right) + N_{84} \,\varphi_{0x} \left( t \right) + N_{85} \,\varphi_{0y} \left( t \right) + N_{86} \,\psi_{0x} \left( t \right) + N_{87} \,\psi_{0x} \left( t \right) + N_{88} \,\phi_{0} \left( t \right)} \right)} } = 0 $$

(116)

in which $N_{ij}$ are calculated as:

$$ \left( {N_{11} ,N_{12} ,N_{13} ,N_{14} ,N_{15} ,N_{16} ,N_{17} ,N_{18} } \right) = \int\limits_{0}^{a + F\left( y \right)} {\int\limits_{0}^{b} {\left( {L_{11} ,L_{12} ,L_{13} ,L_{14} ,L_{15} ,L_{16} ,L_{17} ,L_{18} } \right)} } cos\left( {\frac{m\,\pi x}{{a + F\left( y \right)}}} \right)\sin \left( {\frac{n\,\pi y}{b}} \right), $$

(117)

$$ \left( {N_{21} ,N_{22} ,N_{23} ,N_{24} ,N_{25} ,N_{26} ,N_{27} ,N_{28} } \right) = \int\limits_{0}^{a + F\left( y \right)} {\int\limits_{0}^{b} {\left( {L_{21} ,L_{22} ,L_{23} ,L_{24} ,L_{25} ,L_{26} ,L_{27} ,L_{28} } \right)} } \sin \left( {\frac{m\,\pi x}{{a + F\left( y \right)}}} \right)\cos \left( {\frac{n\,\pi y}{b}} \right), $$

(118)

$$ \left( {N_{31} ,N_{32} ,N_{33} ,N_{34} ,N_{35} ,N_{36} ,N_{37} ,N_{38} } \right) = \int\limits_{0}^{a + F\left( y \right)} {\int\limits_{0}^{b} {\left( {L_{31} ,L_{32} ,L_{33} ,L_{34} ,L_{35} ,L_{36} ,L_{37} ,L_{38} } \right)} } \sin \left( {\frac{m\,\pi x}{{a + F\left( y \right)}}} \right)\sin \left( {\frac{n\,\pi y}{b}} \right), $$

(119)

$$ \left( {N_{41} ,N_{42} ,N_{43} ,N_{44} ,N_{45} ,N_{46} ,N_{47} ,N_{48} } \right) = \int\limits_{0}^{a + F\left( y \right)} {\int\limits_{0}^{b} {\left( {L_{41} ,L_{42} ,L_{43} ,L_{44} ,L_{45} ,L_{46} ,L_{47} ,L_{48} } \right)} } \cos \left( {\frac{m\,\pi x}{{a + F\left( y \right)}}} \right)\sin \left( {\frac{n\,\pi y}{b}} \right), $$

(120)

$$ \left( {N_{51} ,N_{52} ,N_{53} ,N_{54} ,N_{55} ,N_{56} ,N_{57} ,N_{58} } \right) = \int\limits_{0}^{a + F\left( y \right)} {\int\limits_{0}^{b} {\left( {L_{51} ,L_{52} ,L_{53} ,L_{54} ,L_{55} ,L_{56} ,L_{57} ,L_{58} } \right)} } \sin \left( {\frac{m\,\pi x}{{a + F\left( y \right)}}} \right)\cos \left( {\frac{n\,\pi y}{b}} \right), $$

(121)

$$ \left( {N_{61} ,N_{62} ,N_{63} ,N_{64} ,N_{65} ,N_{66} ,N_{67} ,N_{68} } \right) = \int\limits_{0}^{a + F\left( y \right)} {\int\limits_{0}^{b} {\left( {L_{61} ,L_{62} ,L_{63} ,L_{64} ,L_{65} ,L_{66} ,L_{67} ,L_{68} } \right)} } cos\left( {\frac{m\,\pi x}{{a + F\left( y \right)}}} \right)\sin \left( {\frac{n\,\pi y}{b}} \right), $$

(122)

$$ \left( {N_{71} ,N_{72} ,N_{73} ,N_{74} ,N_{75} ,N_{76} ,N_{77} ,N_{78} } \right) = \int\limits_{0}^{a + F\left( y \right)} {\int\limits_{0}^{b} {\left( {L_{71} ,L_{72} ,L_{73} ,L_{74} ,L_{75} ,L_{76} ,L_{77} ,L_{78} } \right)} } sin\left( {\frac{m\,\pi x}{{a + F\left( y \right)}}} \right)\cos \left( {\frac{n\,\pi y}{b}} \right), $$

(123)

$$ \left( {N_{81} ,N_{82} ,N_{83} ,N_{84} ,N_{85} ,N_{86} ,N_{87} ,N_{88} } \right) = \int\limits_{0}^{a + F\left( y \right)} {\int\limits_{0}^{b} {\left( {L_{81} ,L_{82} ,L_{83} ,L_{84} ,L_{85} ,L_{86} ,L_{87} ,L_{88} } \right)} } sin\left( {\frac{m\,\pi x}{{a + F\left( y \right)}}} \right)sin\left( {\frac{n\,\pi y}{b}} \right), $$

(124)