Free vibration and buckling analyses of FG porous sandwich curved microbeams in thermal environment under magnetic field based on modified couple stress theory

Abstract

Porous sandwich structures include different numbers of layers and are capable of demonstrating higher values of strength to weight ratio in comparison with traditional sandwich structures. Free vibration and mechanical buckling responses of a three-layered curved microbeam was investigated under the Lorentz magnetic load in the current study. A viscoelastic substrate was considered and the effect of the thermal environment on its mechanical properties was assessed. The core was composed of the functionally graded porous materials whose properties changed across the thickness based on some given functions. The face sheets were FG-carbon nanotube-reinforced composites and the influence of the placement of CNTs was evaluated on the behavior of the faces. Using the extended rule of mixture, their effective properties were determined. Modified couple stress theory was used to predict the results in the micro-dimension. While the governing equations were derived based on the higher order shear deformation theory and energy method, and mathematically solved via Navier’s method. The results were validated with the previously published works, considering the effects of various parameters. As comprehensively explained in the results section, natural frequencies and critical buckling loads were reduced by enhancing the central opening angle. Moreover, an increase in the porosity coefficient declined the mentioned values, but increasing the CNTs content showed the opposite effect. The outcomes of this study may help in the design and manufacturing of various equipment using such smart structures, making high stiffness to weight ratios more accessible.

Appendix

The non-zero components of stiffness, damping, and mass matrices in Eq. (49) can be defined as:

$$ K_{11} = Q_{110} \Xi^{2} + \frac{{Q_{550} }}{{R^{2} }} + {\mkern 1mu} \frac{{l^{2} \beta_{1} {\mkern 1mu} \Xi^{2} }}{{8R^{2} }}, $$

$$ K_{12} = - Q_{111} \Xi^{3} - \frac{{Q_{110} \Xi }}{R} - \frac{{Q_{551} \Xi }}{{R^{2} }} - {\mkern 1mu} \frac{{l^{2} \beta_{1} {\mkern 1mu} \Xi^{3} }}{4R} - \frac{{l^{2} \beta_{2} {\mkern 1mu} \Xi^{3} }}{{8R^{2} }}, $$

$$ K_{13} = Q_{113} \Xi^{2} - \frac{{Q_{556} }}{R} + \frac{{Q_{553} }}{{R^{2} }} + \frac{{l^{2} \beta_{7} {\mkern 1mu} \Xi^{2} }}{8R} + \frac{{l^{2} \beta_{4} {\mkern 1mu} \Xi^{2} }}{{8R^{2} }}, $$

$$ K_{21} = - Q_{111} \Xi^{3} - \frac{{Q_{110} \Xi }}{R} - \frac{{Q_{551} \Xi }}{{R^{2} }} - \frac{{l^{2} \beta_{1} {\mkern 1mu} \Xi^{3} }}{4R} - \frac{{l^{2} \beta_{2} {\mkern 1mu} \Xi^{3} }}{{8R^{2} }}, $$

$$ \begin{gathered} K_{22} = \frac{{Q_{110} }}{{R^{2} }} + Q_{112} \Xi^{4} - N_{T} \Xi^{2} - N_{0} \Xi^{2} - K_{2} \Xi^{2} - K_{1} + \frac{{Q_{552} \Xi^{2} }}{{R^{2} }} - \eta {\mkern 1mu} H_{x}^{2} \Xi^{2} + 2{\mkern 1mu} \frac{{Q_{111} \Xi^{2} }}{R} + \frac{1}{2}{\mkern 1mu} l^{2} \beta_{1} {\mkern 1mu} \Xi^{4} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\, + \frac{{l^{2} \beta_{1} {\mkern 1mu} \Xi^{2} }}{{8R^{2} }}\, + {\mkern 1mu} \frac{{l^{2} \beta_{2} {\mkern 1mu} \Xi^{4} }}{2R} + {\mkern 1mu} \frac{{l^{2} \beta_{3} {\mkern 1mu} \Xi^{4} }}{{8R^{2} }}, \hfill \\ \end{gathered} $$

$$ \begin{gathered} K_{23} = - Q_{115} \Xi^{3} - \frac{{Q_{113} \Xi }}{R} + \frac{{Q_{557} \Xi }}{R} - \frac{{Q_{555} \Xi }}{{R^{2} }} - \frac{1}{4}{\mkern 1mu} l^{2} \beta_{7} {\mkern 1mu} \Xi^{3} - {\mkern 1mu} \frac{{l^{2} \beta_{6} {\mkern 1mu} \Xi^{3} }}{{8R^{2} }} - {\mkern 1mu} \frac{{l^{2} \beta_{11} {\mkern 1mu} \Xi }}{8R} - \frac{{l^{2} \beta_{7} {\mkern 1mu} \Xi }}{{8R^{2} }} - \frac{{l^{2} \beta_{4} {\mkern 1mu} \Xi^{3} }}{4R} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - {\mkern 1mu} \frac{{l^{2} \beta_{8} {\mkern 1mu} \Xi^{3} }}{8R}, \hfill \\ \end{gathered} $$

$$ K_{31} = Q_{113} \Xi^{2} - \frac{{Q_{556} }}{R} + \frac{{Q_{553} }}{{R^{2} }} + \frac{{l^{2} \beta_{7} {\mkern 1mu} \Xi^{2} }}{8R} + {\mkern 1mu} \frac{{l^{2} \beta_{4} {\mkern 1mu} \Xi^{2} }}{{8R^{2} }}, $$

$$ \begin{gathered} K_{32} = - Q_{115} \Xi^{3} + \frac{{Q_{557} \Xi }}{R} - \frac{1}{4}{\mkern 1mu} l^{2} \beta_{7} {\mkern 1mu} \Xi^{3} - \frac{{Q_{555} \Xi }}{{R^{2} }} - \frac{{Q_{113} \Xi }}{R} - {\mkern 1mu} \frac{{l^{2} \beta_{4} {\mkern 1mu} \Xi^{3} }}{4R} - \frac{{l^{2} \beta_{6} {\mkern 1mu} \Xi^{3} }}{{8R^{2} }} + \frac{{l^{2} \beta_{7} {\mkern 1mu} \Xi }}{{8R^{2} }}\, - {\mkern 1mu} \frac{{l^{2} \beta_{8} {\mkern 1mu} \Xi^{3} }}{8R} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - {\mkern 1mu} \frac{{l^{2} \beta_{11} {\mkern 1mu} \Xi }}{8R}, \hfill \\ \end{gathered} $$

$$ K_{33} = Q_{559} + \frac{1}{8}{\mkern 1mu} l^{2} \beta_{13} + \frac{{Q_{554} }}{{R^{2} }} + Q_{114} \Xi^{2} - 2{\mkern 1mu} \frac{{Q_{558} }}{R} - {\mkern 1mu} \frac{{l^{2} \beta_{10} }}{{8R^{2} }} + \frac{1}{8}l^{2} \beta_{10} {\mkern 1mu} \Xi^{2} + {\mkern 1mu} \frac{{l^{2} \beta_{5} {\mkern 1mu} \Xi^{2} }}{{8R^{2} }} + {\mkern 1mu} \frac{{l^{2} \beta_{9} {\mkern 1mu} \Xi^{2} }}{4R}, $$

$$ M_{11} = - I_{0} {\mkern 1mu} ,\quad \quad \quad M_{12} = I_{1} {\mkern 1mu} \Xi {\mkern 1mu} ,\quad \quad \quad \quad \quad M_{13} = - I_{3} {\mkern 1mu} , $$

$$ M_{21} = I_{1} {\mkern 1mu} \Xi {\mkern 1mu} ,\quad \quad \quad M_{22} = - I_{{0{\mkern 1mu} }} - I_{2} {\mkern 1mu} \Xi^{2} ,\quad \quad \;M_{23} = I_{5} {\mkern 1mu} \Xi , $$

$$ M_{31} = - I_{{3{\mkern 1mu} }} ,\quad \quad \quad M_{32} = I_{5} {\mkern 1mu} \Xi {\mkern 1mu} ,\quad \quad \quad \quad \quad M_{33} = - I_{4} , $$

$$ C_{22} = - D $$