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Vibrational behavior of truncated conical porous GPL-reinforced sandwich micro/nano-shells

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Abstract

In this paper, dynamic characteristics of microconical sandwich shells are investigated. The microshell is considered to consist of a porous core made of a polymer and two face sheets made of that polymer which is reinforced by graphene nanoplatelets (GPLs). The face sheets effective mechanical characteristics are estimated utilizing the rule of mixture along with the Halpin–Tsai model, and size effects are incorporated based on the modified couple stress theory. Hamilton’s principle is applied for derivation of governing equations of motion as well as boundary condition in which differential quadrature method is employed for numerical solution. The accuracy of the presented solution is examined using the benchmark results reported in other papers. Influences of different parameters on the natural frequencies in the various vibrational modes of the microshells are examined, including the wave number, micro-length-scale parameter, the thickness of the porous core, dispersion patterns of the pores and the GPLs, porosity parameter, total mass fraction of the GPLs, and also the semi-vertex angle of the cone.

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Author information

Authors and Affiliations

  1. Department of Solid Mechanic, Faculty of Mechanical Engineering, University of Kashan, Kashan, 87317-51167, Iran

    Niloofar Adab & Mohammad Arefi

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  1. Niloofar Adab
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  2. Mohammad Arefi
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Correspondence to Mohammad Arefi.

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Appendices

Appendix 1

The demonstration of stiffness and mass matrices and displacement vector are as follows:

$$\left\{ s \right\} = \left\{ {\begin{array}{*{20}c} {\left\{ U \right\}} \\ {\left\{ V \right\}} \\ {\left\{ W \right\}} \\ {\left\{ X \right\}} \\ {\left\{ \Theta \right\}} \\ \end{array} } \right\},\quad \left[ K \right] = \left[ {\begin{array}{*{20}c} {k_{11} } & \ldots & {k_{15} } \\ \vdots & \ddots & \vdots \\ {k_{51} } & \ldots & {k_{55} } \\ \end{array} } \right],\quad \left[ M \right] = - \left[ {\begin{array}{*{20}c} {I_{0} I} & {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{1} I} & {\left[ 0 \right]} \\ {\left[ 0 \right]} & {I_{0} I} & {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{1} I} \\ {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{0} I} & {\left[ 0 \right]} & {\left[ 0 \right]} \\ {I_{1} I} & {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{2} I} & {\left[ 0 \right]} \\ {\left[ 0 \right]} & {I_{1} I} & {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{2} I} \\ \end{array} } \right],$$
(47)

in which [0] and I are the zero and the identity matrix of order N and kij are defined accordingly

$$\begin{aligned} k_{11} & = \left( {A_{11} I + 0.25n^{2} a_{{m}} \left[ {a_{2} } \right]} \right)\left[ B \right] + \sin \alpha \left( {A_{11} \left[ {a_{1} } \right] - 0.25n^{2} a_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] \\ & \quad - \left\{ {\left( {A_{22} \sin^{2} \alpha + n^{2} A_{66} } \right)\left[ {a_{2} } \right] + 0.25n^{2} a_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{12} & = 0.25na_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5na_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + n\left\{ {\left( {A_{12} + A_{66} } \right)\left[ {a_{1} } \right] - 0.25a_{{m}} \left( {1 + n^{2} + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - n\sin \alpha \left\{ {\left( {A_{22} + A_{66} } \right)\left[ {a_{2} } \right] + 0.25a_{{m}} \left( {n^{2} - \sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{13} & = \cos \alpha \left( {A_{12} \left[ {a_{1} } \right] - 0.75n^{2} a_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] - 0.5\sin 2\alpha \left( {A_{22} \left[ {a_{2} } \right] - 0.5n^{2} a_{{m}} \left[ {a_{4} } \right]} \right) \\ k_{14} & = \left( {B_{11} I + 0.25n^{2} b_{{m}} \left[ {a_{2} } \right]} \right)\left[ B \right] + \sin \alpha \left( {B_{11} \left[ {a_{1} } \right] - 0.25n^{2} b_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] \\ & \quad - \left\{ {\left( {B_{22} \sin^{2} \alpha + n^{2} B_{66} } \right)\left[ {a_{2} } \right] - 0.75n^{2} a_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right.\left. { + 0.25n^{2} b_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{15} & = 0.25nb_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nb_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + n\left\{ {\left( {B_{12} + B_{66} } \right)\left[ {a_{1} } \right] - 0.25b_{{m}} \left( {1 + n^{2} + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - n\sin \alpha \left\{ {\left( {B_{22} + B_{66} } \right)\left[ {a_{2} } \right] - 0.5a_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.25b_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ \end{aligned}$$
$$\begin{aligned} k_{21} & = - 0.25na_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5na_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + \left\{ { - n\left( {A_{12} + A_{66} } \right)\left[ {a_{1} } \right] + 0.25na_{{m}} \left( {n^{2} + 2 - 5\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - n\sin \alpha \left\{ {\left( {A_{22} + A_{66} } \right)\left[ {a_{2} } \right] + 0.25a_{{m}} \left( {2 - 5\sin^{2} \alpha + 3n^{2} } \right)\left[ {a_{4} } \right]} \right\} \\ k_{22} & = - 0.25a_{{m}} \left[ D \right] - 0.5a_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left\{ {A_{66} I + 0.25a_{{m}} \left( {2 - n + \sin^{2} \alpha } \right)\left[ {a_{2} } \right]} \right\}\left[ B \right] \\ & \quad + \sin \alpha \left\{ {A_{66} \left[ {a_{1} } \right] - 0.25a_{{m}} \left( {2 + n^{2} + \sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - \left\{ {\left( {A_{44} \cos^{2} \alpha + A_{66} \sin^{2} \alpha + n^{2} A_{22} } \right)\left[ {a_{2} } \right] + 0.25a_{{m}} \left[ {n^{2} \left( {1 + 2\sin^{2} \alpha } \right) - 3\sin^{4} \alpha } \right]\left[ {a_{4} } \right]} \right\} \\ k_{23} & = 0.5na_{{m}} \cos \alpha \left[ {a_{2} } \right]\left[ B \right] - 0.625na_{{m}} \sin 2\alpha \left[ {a_{3} } \right]\left[ A \right] - n\cos \alpha \left\{ {\left( {A_{22} + A_{44} } \right)\left[ {a_{2} } \right]} \right. \\ &\left. { \quad + 0.25a_{{m}} \left( {n^{3} - 4\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{24} & = - 0.25nb_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nb_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] - n\left\{ {\left( {B_{12} + B_{66} } \right)\left[ {a_{1} } \right] + 0.5a_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right. \\ & \quad \left. { - 0.25b_{{m}} \left( {2 + n^{2} - 5\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - n\sin \alpha \left\{ {\left( {B_{22} + B_{66} } \right)\left[ {a_{2} } \right] - 0.25a_{{m}} \cos \alpha \left( {1 + 2\sin \alpha } \right)\left[ {a_{3} } \right]} \right. \\ &\left. { \quad + 0.25b_{{m}} \left[ {\sin \alpha \left( {4\cos^{2} \alpha - 3\sin \alpha } \right) + 3n^{2} } \right]\left[ {a_{4} } \right]} \right\} \\ k_{25} & = - 0.25b_{{m}} \left[ D \right] - 0.5b_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left\{ {B_{66} + 0.25b_{{m}} \left( {2 + n^{2} + \sin^{2} \alpha } \right)\left[ {a_{2} } \right]} \right\}\left[ B \right] \\ & \quad + \sin \alpha \left[ {B_{66} \left[ {a_{1} } \right] - 0.5a_{{m}} \cos \alpha \left[ {a_{2} } \right] - 0.25b_{{m}} \left( {n^{2} + 3\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right]\left[ A \right] + \left\{ {A_{44} \cos \alpha \left[ {a_{1} } \right]} \right. \\ &\left. { \quad - \left( {B_{66} \sin^{2} \alpha + n^{2} B_{22} } \right)\left[ {a_{2} } \right] + 0.25a_{{m}} \cos \alpha \left( {2\sin^{2} \alpha - n^{2} } \right)\left[ {a_{3} } \right] - 0.25b_{{m}} \sin^{2} \alpha \left( {4 + 3n^{2} - 7\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \end{aligned}$$
$$\begin{aligned} k_{31} & = - \cos \alpha \left( {A_{12} \left[ {a_{1} } \right] - 0.75n^{2} a_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] - 0.5\sin 2\alpha \left( {A_{22} \left[ {a_{2} } \right] + n^{2} a_{{m}} \left[ {a_{4} } \right]} \right) \\ k_{32} & = 0.5na_{{m}} \cos \alpha \left[ {a_{2} } \right]\left[ B \right] + 0.25na_{{m}} \sin 2\alpha \left[ {a_{3} } \right]\left[ A \right] - n\cos \alpha \left\{ {\left( {A_{22} + A_{44} } \right)\left[ {a_{2} } \right] + 0.25n^{2} a_{{m}} \cos \alpha \left[ {a_{4} } \right]} \right\} \\ k_{33} & = - 0.25a_{{m}} \left[ D \right] - 0.5a_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left[ {A_{55} I + 0.25a_{{m}} \left( {1 + 2n^{2} } \right)\left[ {a_{2} } \right]} \right]\left[ B \right] \\ & \quad + \sin \alpha \left\{ {A_{55} \left[ {a_{1} } \right] - 0.25a_{{m}} \left[ {\sin \alpha \left( {\sin \alpha + \cos \alpha } \right) + 2n^{2} } \right]\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - \left\{ {\left( {A_{22} \cos^{2} \alpha + n^{2} A_{44} } \right)\left[ {a_{2} } \right] + 0.25n^{2} a_{{m}} \left( {n^{2} - 4\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{34} & = 0.25a_{{m}} \left[ C \right] + 0.5a_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ B \right] + \left\{ {A_{55} I - B_{12} \cos \alpha \left[ {a_{1} } \right] - 0.25a_{{m}} \left( {1 + n^{2} } \right)\left[ {a_{2} } \right]} \right. \\ & \quad \left. { + 0.75n^{2} b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right\}\left[ A \right] + \sin \alpha \left\{ {A_{55} \left[ {a_{1} } \right] - B_{22} \cos \alpha \left[ {a_{2} } \right] + 0.25a_{{m}} \left[ {\sin \alpha \left( {\sin \alpha + \cos \alpha } \right) - n^{2} } \right]\left[ {a_{3} } \right]} \right. \\ &\left. { \quad - n^{2} b_{{m}} \cos \alpha \left[ {a_{4} } \right]} \right\} \\ k_{35} & = 0.25n\left( {a_{{m}} \left[ {a_{1} } \right] + 3b_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right)\left[ B \right] - 0.5n\sin \alpha \left( {0.5a_{{m}} \left[ {a_{2} } \right] - b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right)\left[ A \right] \\ & \quad + n\left\{ {A_{44} \left[ {a_{1} } \right] - B_{22} \cos \alpha \left[ {a_{2} } \right] + 0.25a_{{m}} \left( {\sin^{2} \alpha - n^{2} } \right)\left[ {a_{3} } \right] - 0.5b_{{m}} \sin \alpha \sin 2\alpha \left[ {a_{4} } \right]} \right\} \\ \end{aligned}$$
$$\begin{aligned} k_{41} & = \left( {B_{11} I + 0.25n^{2} b_{{m}} \left[ {a_{2} } \right]} \right)\left[ B \right] + \sin \alpha \left( {B_{11} \left[ {a_{1} } \right] - 0.25n^{2} b_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] - \left\{ {\left( {B_{22} \sin^{2} \alpha + n^{2} B_{66} } \right)\left[ {a_{2} } \right]} \right. \\ &\left. { \quad - 0.75n^{2} a_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.25n^{2} b_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{42} & = 0.25nb_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nb_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right]\, + n\left\{ {\left( {B_{12} + B_{66} } \right)\left[ {a_{1} } \right] + 0.5a_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right. - 0.25b_{{m}} \left( {1 + n^{2} } \right. \\ & \quad \left. {\left. { + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - n\sin \alpha \left\{ {\left( {B_{22} + B_{66} } \right)\left[ {a_{2} } \right] - 0.25a_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.25b_{{m}} \left( {n^{2} - \sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{43} & = - 0.25a_{{m}} \left[ C \right] - 0.25a_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ B \right] + \left\{ { - A_{55} I + B_{12} \cos \alpha \left[ {a_{1} } \right] + 0.25a_{{m}} \left( {1 + n^{2} } \right)\left[ {a_{2} } \right]} \right. \\ & \quad \left. { - 0.75n^{2} b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right\}\left[ A \right] - \sin \alpha \left( {B_{22} \cos \alpha \left[ {a_{2} } \right] + 0.5n^{2} a_{{m}} \left[ {a_{3} } \right] - 0.5n^{2} b_{{m}} \cos \alpha \left[ {a_{4} } \right]} \right) \\ k_{44} & = \left\{ {\left( {D_{11} + 0.25a_{{m}} } \right)I + 0.25n^{2} d_{{m}} \left[ {a_{2} } \right]} \right\}\left[ B \right] + 0.25\sin \alpha \left\{ {\left( {4D_{11} + a_{{m}} } \right)\left[ {a_{1} } \right] - n^{2} d_{{m}} \left[ {a_{3} } \right]} \right\}\left[ A \right] - \left\{ {A_{55} I} \right. \\ & \quad \left. { + \left[ {D_{22} \sin^{2} \alpha + n^{2} D_{66} + a_{{m}} \left( {0.25 + n^{2} } \right)} \right]\left[ {a_{2} } \right] - 1.5n^{2} b_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.25n^{2} d_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{45} & = 0.25nd_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nd_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + n\left\{ {\left( {D_{12} + D_{66} - 0.75a_{{m}} } \right)\left[ {a_{1} } \right] + 0.75b_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right. \\ & \quad \left. { - 0.25d_{{m}} \left( {1 + n^{2} + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - n\sin \alpha \left\{ {\left( {D_{22} + D_{66} + 1.25a_{{m}} } \right)\left[ {a_{2} } \right] - 1.25b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right. \\ & \quad \left. { + 0.25d_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ \end{aligned}$$
$$\begin{aligned} k_{51} & = - 0.25nb_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nb_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + n\left\{ { - \left( {B_{12} + B_{66} } \right)\left[ {a_{1} } \right]} \right. \\ & \quad \left. { + 0.25b_{{m}} \left( {2 + n^{2} - 5\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - n\sin \alpha \left\{ {2I_{1} \Omega^{2} I + \left( {B_{22} + B_{66} } \right)\left[ {a_{2} } \right]} \right. \\ & \quad \left. { - 0.5a_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.75b_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{52} & = - 0.25b_{{m}} \left[ D \right] - 0.5b_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left\{ {B_{66} I + 0.5a_{{m}} \cos \alpha \left[ {a_{1} } \right]\, + 0.25b_{{m}} \left( {2 + n^{2} + \sin^{2} \alpha } \right)\left[ {a_{2} } \right]} \right\}\left[ B \right] \\ & \quad + \sin \alpha \left\{ {B_{66} \left[ {a_{1} } \right] - 0.25b_{{m}} \left( {3 + n^{2} + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - \left\{ { - A_{44} \cos \alpha \left[ {a_{1} } \right] + \left( {B_{66} \sin^{2} \alpha + n^{2} B_{22} } \right)\left[ {a_{2} } \right]} \right. \\ & \quad \left. { + 0.25a_{{m}} \cos \alpha \left( {n^{2} - 2\sin^{2} \alpha } \right)\left[ {a_{3} } \right] + 0.75b_{{m}} \sin^{2} \alpha \left( {n^{2} - \sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{53} & = - 0.25n\left( {a_{{m}} \left[ {a_{1} } \right] + 3b_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right)\left[ B \right] - n\sin \alpha \left( { - 0.25a_{{m}} \left[ {a_{2} } \right] + 2b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right)\left[ A \right] \\ & \quad - n\left\{ { - A_{44} \left[ {a_{1} } \right] + B_{22} \cos \alpha \left[ {a_{2} } \right] + 0.25n^{2} a_{{m}} \left[ {a_{3} } \right] - 0.75b_{{m}} \sin \alpha \sin 2\alpha \left[ {a_{4} } \right]} \right\} \\ k_{54} & = - 0.25nd_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nd_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] - n\left\{ {\left( {D_{12} + D_{66} } \right)\left[ {a_{1} } \right] - 0.75a_{{m}} \left[ {a_{1} } \right]} \right. \\ & \quad \left. { + 0.75b_{{m}} \cos \alpha \left[ {a_{2} } \right] - 0.25d_{{m}} \left( {2 + n^{2} - 5\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - n\sin \alpha \left\{ {\left( {D_{22} + D_{66} + 1.25a_{{m}} } \right)\left[ {a_{2} } \right] - 2b_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.75d_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{55} & = - 0.25d_{{m}} \left[ D \right] - 0.5d_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left\{ {\left( {D_{66} + a_{{m}} } \right)I + 0.25d_{{m}} \left( {2 + n^{2} + \sin^{2} \alpha } \right)\left[ {a_{2} } \right]} \right\}\left[ B \right] \\ & \quad - 0.25n^{2} d_{{m}} \sin \alpha \left[ {a_{3} } \right]\left[ A \right] + n\sin \alpha \left\{ {\left( {D_{66} + a_{{m}} } \right)\left[ {a_{1} } \right] - 0.25d_{{m}} \left( {2 + \sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - \left\{ {A_{44} I} \right. \\ & \quad + \left[ {\left( {D_{66} + a_{{m}} } \right)\sin^{2} \alpha + n^{2} \left( {D_{22} + 0.25a_{{m}} } \right)} \right]\left[ {a_{2} } \right] - 0.75b_{{m}} \sin \alpha \sin 2\alpha \left[ {a_{3} } \right] \\ & \quad \left. { + 0.75d_{{m}} \sin^{2} \alpha \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ \end{aligned}$$
(48)

and with the following definitions, [a1]–[a4] are four diagonal matrices:

$$\left[ {a_{1} } \right]_{ii} = r_{i}^{ - 1} ,\quad \left[ {a_{2} } \right]_{ii} = r_{i}^{ - 2} ,\quad \left[ {a_{3} } \right]_{ii} = r_{i}^{ - 3} ,\quad \left[ {a_{4} } \right]_{ii} = r_{i}^{ - 4} .$$
(49)

Appendix 2

The general form of matrix [P] is defined as follows:

$$\left[ P \right] = \left[ {\begin{array}{*{20}c} {p_{11} } & {p_{12} } & {p_{13} } & {p_{14} } & {p_{15} } \\ {p_{21} } & {p_{22} } & {p_{23} } & {p_{24} } & {p_{25} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {p_{161} } & {p_{162} } & {p_{163} } & {p_{164} } & {p_{165} } \\ \end{array} } \right],$$
(50)

in which p11–p85 are related to the conditions at x = 0 and p81–p165 are related to the conditions at x = L. As an example, for a microshell clamped (C) at x = 0 and simply supported (S) at x = L (abbreviated as CS), p11–p165 can be stated as follows:

$$\begin{aligned} &p_{11} = p_{22} = p_{33} = p_{44} = p_{55} = I_{1} ,\quad {p_{62} = p_{73} = p_{85} = A_{1} ,} \\ & p_{12} = p_{13} = p_{14} = p_{15} = p_{21} = p_{23} = p_{24} = p_{25} = p_{31} = p_{32} = p_{34} = p_{35} = p_{41} = p_{42} = p_{43} = p_{45} = p_{51} \\ & \quad = p_{52} = p_{53} = p_{54} = p_{61} = p_{63} = p_{64} = p_{65} = p_{71} = p_{72} = p_{74} = p_{75} = p_{81} = p_{82} = p_{83} = p_{84} = \left\{ 0 \right\}_{1 \times N} , \end{aligned}$$
(51)
$$\begin{aligned} & p_{91} = A_{11} A_{N} + \frac{{A_{12} \sin \alpha }}{b}I_{N} ,\quad {p_{94} = p_{121} = B_{11} A_{N} + \frac{{B_{12} \sin \alpha }}{b}I_{N} ,} \quad {p_{124} = D_{11} A_{N} + \frac{{D_{12} \sin \alpha }}{b}I_{N} ,} \\ & p_{102} = p_{113} = p_{135} = I_{N} , \\ & p_{141} = \frac{{na_{{m}} }}{b}A_{N} - \frac{{na_{{m}} \sin \alpha }}{{b^{2} }}I_{N} , \quad {p_{142} = a_{{m}} B_{N} + \frac{{a_{{m}} \sin \alpha }}{b}A_{N} ,} \quad {p_{144} = \frac{{nb_{{m}} }}{b}A_{N} - \frac{{nb_{{m}} \sin \alpha }}{{b^{2} }}I_{N} ,} \\ & p_{145} = b_{{m}} B_{N} + \frac{{b_{{m}} \sin \alpha }}{b}A_{N} , \quad {p_{153} = - a_{{m}} B_{N} + \frac{{a_{{m}} \sin \alpha }}{b}A_{N} ,} \quad {p_{154} = a_{{m}} A_{N} - \frac{{a_{{m}} \sin \alpha }}{b}I_{N} ,} \\ & p_{161} = \frac{{nb_{{m}} }}{r}A_{N} - \frac{{nb_{{m}} \sin \alpha }}{{r^{2} }}I_{N} , \quad {p_{162} = b_{{m}} B_{N} + \frac{{b_{{m}} \sin \alpha }}{b}A_{N} ,} \quad {p_{164} = \frac{{nd_{{m}} }}{b}A_{N} - \frac{{nd_{{m}} \sin \alpha }}{{b^{2} }}I_{N} ,} \\ \\ & p_{165} = d_{{m}} B_{N} + \frac{{d_{{m}} \sin \alpha }}{b}A_{N} , \\ & p_{92} = p_{93} = p_{95} = p_{101} = p_{103} = p_{104} = p_{105} = p_{111} = p_{112} = p_{114} = p_{115} = p_{122} \\ & \quad = p_{123} = p_{125} = p_{131} = p_{132} = p_{133} = p_{134} = p_{143} = p_{151} = p_{152} = p_{155} = p_{163} = \left\{ 0 \right\}_{1 \times N} , \\ \end{aligned}$$

in which the subscripts 1 and N, respectively, indicate the first and Nth row of the matrices.

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Adab, N., Arefi, M. Vibrational behavior of truncated conical porous GPL-reinforced sandwich micro/nano-shells. Engineering with Computers 39, 419–443 (2023). https://doi.org/10.1007/s00366-021-01580-8

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