Bi-directional thermal buckling and resonance frequency characteristics of a GNP-reinforced composite nanostructure

Abstract

In this article, thermal buckling and resonance frequency of a composite cylindrical nanoshell reinforced with graphene nanoplatelets (GNP) under bi-directional thermal loading are presented. The temperature-dependent material properties of piece-wise GNP-reinforced composites (GNPRC) are assumed to be graded in the thickness direction of a cylindrical nanoshell. Also, Halphin-Tsai nanomechanical model is used to surmise the effective material properties of each layer. The size-dependent GNPRC nanoshell is analyzed using modified couple stress parameter (FMCS). For the first time, in the presented study show that bi-directional thermal buckling occurs if the percent of relative frequency change tends to 30%. The novelty of the current study is in considering the effects of bi-directional thermal loading in addition of FMCS on relative frequency, resonance frequencies, thermal buckling, and dynamic deflection of the GNPRC nanoshell. The governing equations and boundary conditions are developed using Hamilton’s principle and solved with the aid of analytical method. The results show that, various bi-directionasl thermal loading and other geometrical and mechanical properties have important role on resonance frequency, relative frequency change, thermal buckling, and dynamic deflection of the GNPRC cylindrical nanoshell. The results of the current study are useful suggestions for design of materials science, micro-mechanical and nano-mechanical systems such as microactuators and microsensors.

Appendices

Appendix A

The components of the matrices in Eq. (24):

$$K_{11} = \left\{ \begin{gathered} A_{11} ( - m^{2} ) - A_{66} \left( \frac{n}{R} \right)^{2} + \frac{{A_{77} l^{2} n^{2} }}{{4R^{2} }}( - m^{2} ) \hfill \\ + \left( { - \frac{{A_{77} l^{2} n^{4} }}{{4R^{4} }}} \right) - \frac{{A_{77} l^{2} n^{2} }}{{R^{4} }} \hfill \\ \end{gathered} \right\}$$

$$K_{12} = \left\{ \begin{gathered} \frac{{A_{12} n}}{R}(m) + \frac{{A_{66} n}}{R}(m) + \frac{{A_{77} l^{2} n}}{4R}( - m^{3} ) \hfill \\ + \left( {\frac{{A_{77} l^{2} ( - n - n^{3} )}}{{4R^{3} }}} \right)(m) \hfill \\ \end{gathered} \right\}$$

$$K_{13} = \left\{ {\frac{{A_{12} }}{R}(m) - \left( {\frac{{A_{77} l^{2} n^{2} }}{{2R^{3} }}} \right)(m)} \right\}$$

$$K_{14} = \left\{ \begin{gathered} B_{11} ( - m^{2} ) - B_{66} \frac{{n^{2} }}{{R^{2} }} + \frac{{B_{77} l^{2} n^{2} }}{{4R^{2} }}( - m^{2} ) \hfill \\ + \left( {\frac{{5A_{77} l^{2} n^{2} }}{{4R^{3} }} - \frac{{B_{77} l^{2} n^{4} }}{{4R^{4} }}} \right) \hfill \\ \end{gathered} \right\}$$

$$K_{15} = \left\{ \begin{gathered} \frac{{B_{12} n}}{R}(m) + \frac{{B_{66} n}}{R}(m) + \frac{{B_{77} l^{2} n}}{4R}( - m^{3} ) \hfill \\ - \left( {\frac{{(B_{77} )l^{2} n}}{{2R^{3} }} + \frac{{A_{77} l^{2} n}}{{4R^{2} }} - \frac{{B_{77} l^{2} n^{3} }}{{4R^{3} }}} \right)(m) \hfill \\ \end{gathered} \right\}$$

$$K_{21} = - \left\{ \begin{gathered} + \frac{{A_{12} n}}{R}( - m) + \frac{{A_{66} n}}{R}( - m) + \frac{{A_{77} l^{2} n}}{4R}(m^{3} ) \hfill \\ + \left( {\frac{{A_{77} l^{2} ( - n - n^{3} )}}{{4R^{3} }}} \right)( - m) \hfill \\ \end{gathered} \right\}$$

$$K_{22} = \left\{ \begin{gathered} A_{66} \sum\limits_{k = 1}^{i} {( - m^{2} )} - A_{11} \left( \frac{n}{R} \right)^{2} - \frac{{k_{s} A_{66} }}{{R^{2} }} - \frac{{A_{77} l^{2} }}{4}(m^{4} ) \hfill \\ + \left( {\frac{{A_{77} l^{2} (2 + n^{2} )}}{{4R^{2} }}} \right)( - m^{2} ) \hfill \\ + \left( { - \frac{{A_{77} l^{2} (1 + n^{2} )}}{{4R^{4} }}} \right) - N_{h} \frac{n}{{R^{2} }} \hfill \\ \end{gathered} \right\}$$

$$K_{23} = \left\{ \begin{gathered} - A_{11} \frac{n}{{R^{2} }} - k_{s} A_{66} \frac{n}{{R^{2} }} + \left( {\frac{{A_{66} l^{2} n}}{{4R^{2} }}} \right)( - m^{2} ) \hfill \\ + \left( { - \frac{{A_{566} l^{2} (n^{3} + n)}}{{4R^{4} }}} \right) \hfill \\ \end{gathered} \right\}$$

$$K_{24} = \left\{ \begin{gathered} - \frac{{B_{66} n}}{R}( - m) - \frac{{B_{12} n}}{R}( - m) - \frac{{B_{77} l^{2} n}}{4R}(m^{3} ) \hfill \\ - \left( {\frac{{A_{77} l^{2} n}}{{2R^{2} }} + \frac{{B_{77} l^{2} ( - n^{3} + n)}}{{4R^{3} }}} \right)( - m) \hfill \\ \end{gathered} \right\}$$

$$K_{25} = \left\{ \begin{gathered} B_{66} ( - m^{2} ) - B_{11} \left( \frac{n}{R} \right)^{2} + \frac{{k_{s} A_{55} }}{R} - \frac{{B_{77} l^{2} }}{4}(m^{4} ) \hfill \\ - \left( {\frac{{ - 3A_{77} l^{2} }}{4R} - \frac{{B_{77} l^{2} }}{2R} - \frac{{B_{77} l^{2} (n^{2} - 1)}}{{4R^{2} }}} \right)( - m^{2} ) \hfill \\ + \left( {\frac{{A_{77} l^{2} (1 - n^{2} )}}{{4R^{3} }}} \right) \hfill \\ \end{gathered} \right\}$$

$$K_{31} = - \left\{ { + \frac{{A_{12} }}{R}( - m) - \left( {\frac{{A_{77} l^{2} n^{2} }}{{2R^{3} }}} \right)( - m)} \right\}$$

$$K_{32} = \left\{ \begin{gathered} - A_{11} \frac{n}{{R^{2} }} - k_{s} A_{66} \frac{n}{{R^{2} }} + \left( {\frac{{A_{77} l^{2} n}}{{4R^{2} }}} \right)( - m^{2} ) \hfill \\ + \left( { - \frac{{A_{77} l^{2} (n^{3} + n)}}{{4R^{4} }}} \right) \hfill \\ \end{gathered} \right\}$$

$$K_{33} = \left\{ \begin{gathered} k_{s} A_{55} ( - m^{2} ) - k_{s} A_{66} \left( \frac{n}{R} \right)^{2} - \frac{{A_{11} }}{{R^{2} }} - \frac{{A_{77} l^{2} }}{4}(m^{4} ) + \hfill \\ \left( {\frac{{A_{77} l^{2} (2n^{2} + 1)}}{{4R^{2} }}} \right)( - m^{2} ) \hfill \\ \left( { - \frac{{A_{77} l^{2} (n^{2} + n^{4} )}}{{4R^{4} }}} \right) - N_{h} \frac{{n^{2} }}{{R^{2} }} \hfill \\ \end{gathered} \right\}$$

$$K_{34} = \left\{ \begin{gathered} k_{s} A_{66} ( - m) - \frac{{B_{12} }}{R}( - m) + \frac{{A_{77} l^{2} }}{4}(m^{3} ) \hfill \\ - \left( {\frac{{A_{77} l^{2} (n^{2} + 1)}}{{4R^{2} }}} \right)( - m) \hfill \\ \end{gathered} \right\}$$

$$K_{35} = \left\{ \begin{gathered} \frac{{k_{s} A_{55} n}}{R} - \frac{{B_{11} n}}{{R^{2} }} + \left( {\frac{{B_{77} l^{2} n}}{{2R^{2} }} - \frac{{A_{77} l^{2} n}}{4R}} \right)( - m^{2} ) \hfill \\ + \left( {\frac{{A_{77} l^{2} (n - n^{3} )}}{{4R^{3} }}} \right) \hfill \\ \end{gathered} \right\}$$

$$K_{41} = \left\{ \begin{gathered} + B_{11} ( - m^{2} ) - B_{66} \frac{{n^{2} }}{{R^{2} }} + \frac{{B_{77} l^{2} n^{2} }}{{4R^{2} }}( - m^{2} ) \hfill \\ + \left( {\frac{{5A_{77} l^{2} n^{2} }}{{4R^{3} }} - \frac{{B_{77} l^{2} n^{4} }}{{4R^{4} }}} \right) \hfill \\ \end{gathered} \right\}$$

$$K_{42} = - \left\{ \begin{gathered} - \frac{{B_{66} n}}{R}(m) - \frac{{B_{12} n}}{R}(m) - \frac{{B_{77} l^{2} n}}{4R}( - m^{3} ) \hfill \\ - \left( {\frac{{A_{77} l^{2} n}}{{2R^{2} }} + \frac{{B_{77} l^{2} ( - n^{3} + n)}}{{4R^{3} }}} \right)(m) \hfill \\ \end{gathered} \right\}$$

$$K_{43} = - \left\{ \begin{gathered} k_{s} A_{66} (m) - \frac{{B_{12} }}{R}(m) + \frac{{A_{77} l^{2} }}{4}( - m^{3} ) \hfill \\ - \left( {\frac{{A_{77} l^{2} (n^{2} + 1)}}{{4R^{2} }}} \right)(m) \hfill \\ \end{gathered} \right\}$$

$$K_{44} = \left\{ \begin{gathered} - D_{66} \left( \frac{n}{R} \right)^{2} - k_{s} A_{66} + \left( {\frac{{D_{77} l^{2} n^{2} }}{{4R^{2} }} + \frac{{A_{77} l^{2} }}{4}} \right)( - m^{2} ) \hfill \\ + \left( { - \frac{{D_{77} l^{2} n^{4} }}{{4R^{4} }} + \frac{{2B_{77} l^{2} n^{2} }}{{4R^{3} }} - \frac{{A_{77} l^{2} (1 + 4n^{2} )}}{{4R^{2} }}} \right)D_{11} ( - m^{2} ) \hfill \\ \end{gathered} \right\}$$

$$K_{45} = \left\{ \begin{gathered} \frac{{D_{12} n}}{R}(m) + D_{66} \frac{n}{R}(m) + \frac{{D_{77} l^{2} n}}{4R}( - m^{3} ) \hfill \\ + \left( { - \frac{{D_{77} l^{2} n^{3} }}{{4R^{3} }} + \frac{{B_{77} l^{2} n}}{2R} - \frac{{3A_{77} l^{2} n}}{4R}} \right)(m) \hfill \\ \end{gathered} \right\}$$

$$K_{51} = \left\{ \begin{gathered} + \frac{{B_{12} n}}{R}( - m) + \frac{{B_{66} n}}{R}( - m) + \frac{{B_{77} l^{2} n}}{4R}\sum\limits_{k = 1}^{i} {C_{i,k}^{\left( 3 \right)} } \hfill \\ - \left( {\frac{{(B_{77} )l^{2} n}}{{2R^{3} }} + \frac{{A_{77} l^{2} n}}{{4R^{2} }} - \frac{{B_{77} l^{2} n^{3} }}{{4R^{3} }}} \right)( - m) \hfill \\ \end{gathered} \right\}$$

$$K_{52} = \left\{ \begin{gathered} B_{66} ( - m^{2} ) - B_{11} (\frac{n}{R})^{2} + \frac{{k_{s} A_{66} }}{R} - \frac{{B_{77} l^{2} }}{4}(m^{4} ) \hfill \\ - \left( {\frac{{ - 3A_{77} l^{2} }}{4R} - \frac{{B_{77} l^{2} }}{2R} - \frac{{B_{77} l^{2} (n^{2} - 1)}}{{4R^{2} }}} \right)( - m^{2} ) \hfill \\ + \left( {\frac{{A_{77} l^{2} (1 - n^{2} )}}{{4R^{3} }}} \right) \hfill \\ \end{gathered} \right\}$$

$$K_{53} = \left\{ \begin{gathered} \frac{{k_{s} A_{77} n}}{R} - \frac{{B_{11} n}}{{R^{2} }} + \left( {\frac{{B_{77} l^{2} n}}{{2R^{2} }} - \frac{{A_{77} l^{2} n}}{4R}} \right)( - m^{2} ) \hfill \\ + \left( {\frac{{A_{77} l^{2} (n - n^{3} )}}{{4R^{3} }}} \right) \hfill \\ \end{gathered} \right\}$$

$$K_{54} = - \left\{ \begin{gathered} + \frac{{D_{12} n}}{R}( - m) + D_{66} \frac{n}{R}( - m) + \frac{{D_{77} l^{2} n}}{4R}(m^{3} ) \hfill \\ + \left( { - \frac{{D_{77} l^{2} n^{3} }}{{4R^{3} }} + \frac{{B_{77} l^{2} n}}{2R} - \frac{{3A_{77} l^{2} n}}{4R}} \right)( - m) \hfill \\ \end{gathered} \right\}$$

$$K_{55} = \left\{ \begin{gathered} - D_{11} (\frac{n}{R})^{2} + D_{66} ( - m^{2} ) - k_{s} A_{55} - \frac{{D_{77} l^{2} }}{4}(m^{4} ) \hfill \\ + \left( {\frac{{D_{77} l^{2} n^{2} }}{{4R^{2} }} + \frac{{B_{77} l^{2} }}{2R} + A_{77} l^{2} + \frac{{D_{77} l^{2} }}{2R}} \right)( - m^{2} ) \hfill \\ + \left( { - \frac{{A_{77} l^{2} n^{2} }}{{4R^{2} }} - \frac{{A_{77} l^{2} }}{{4R^{2} }}} \right) \hfill \\ \end{gathered} \right\}$$

$$\begin{gathered} M_{11} = I_{0} ,M_{14} = I_{1} ,M_{22} = I_{0} ,M_{25} = I_{1} ,M_{33} = I_{0} ,M_{41} = I_{1} ,\,M_{44} = I_{2} ,\,M_{52} = I_{1} ,\,M_{55} = I_{2} \hfill \\ \,M_{12} = M_{13} = M_{15} = M_{21} = M_{23} = M_{24} = M_{31} = M_{32} = M_{34} = M_{35} = M_{42} = M_{43} = M_{45} = \,M_{51} = M_{53} = M_{54} = 0\, \hfill \\ \end{gathered}$$

Appendix B

$$B_{1} = \frac{1}{\Delta F}\left| {\begin{array}{*{20}c} {T_{i} } & {K_{o} \left( {q\left( {r_{i} } \right)} \right)} \\ {T_{o} } & {K_{o} \left( {q\left( {r_{o} } \right)} \right)} \\ \end{array} } \right| \, C_{1} = \frac{1}{\Delta F}\left| {\begin{array}{*{20}c} {I_{o} \left( {q\left( {r_{i} } \right)} \right)} & {T_{i} } \\ {I_{o} \left( {q\left( {r_{o} } \right)} \right)} & {T_{o} } \\ \end{array} } \right| \, \Delta F = \left| {\begin{array}{*{20}c} {I_{o} \left( {q\left( {r_{i} } \right)} \right)} & {K_{o} \left( {q\left( {r_{i} } \right)} \right)} \\ {I_{o} \left( {q\left( {r_{o} } \right)} \right)} & {K_{o} \left( {q\left( {r_{o} } \right)} \right)} \\ \end{array} } \right|$$

2.1 FG-V:

$${\text{A}}_{1} = 1 - \frac{{2D_{z} r_{i} }}{h},{\text{ A}}_{2} = \frac{{2D_{z} }}{h},{\text{ A}}_{3} = P_{m}^{2} \left( {\frac{{2D_{x} r_{i} }}{h} - 1} \right),{\text{ A}}_{4} = - 2P_{m}^{2} \frac{{D_{x} }}{h}, \, \frac{{k_{i} }}{{k_{m} }} = 1 + 2D_{i} \left( {\frac{{z - r_{i} }}{h}} \right)$$

2.2 FG-X:

$$\begin{gathered} {\text{when: z }}\langle {\text{ r}}_{i} { + }\frac{h}{2} \hfill \\ {\text{A}}_{1} = 1 + 2D_{z} \left( {1 + \frac{{2r_{i} }}{h}} \right),{\text{ A}}_{2} = - \frac{{4D_{z} }}{h},{\text{ A}}_{3} = - P_{m}^{2} \left( {1 + 2D_{x} \left( {1 + \frac{{2r_{i} }}{h}} \right)} \right),{\text{ A}}_{4} = 4P_{m}^{2} \frac{{D_{x} }}{h} \hfill \\ \frac{{k_{i} }}{{k_{m} }} = 1 + 2D_{i} \left( {1 - \frac{{2\left( {z - r_{i} } \right)}}{h}} \right) \hfill \\ {\text{when: z }}\rangle {\text{ r}}_{i} { + }\frac{h}{2} \hfill \\ {\text{A}}_{1} = 1 - 2D_{z} \left( {1 + \frac{{2r_{o} }}{h}} \right),{\text{ A}}_{2} = \frac{{4D_{z} }}{h},{\text{ A}}_{3} = - P_{m}^{2} \left( {1 - 2D_{x} \left( {1 + \frac{{2r_{o} }}{h}} \right)} \right),{\text{ A}}_{4} = - 4P_{m}^{2} \frac{{D_{x} }}{h} \hfill \\ \frac{{k_{i} }}{{k_{m} }} = 1 + 2D_{i} \left( { - 1 + \frac{{2\left( {z - r_{i} } \right)}}{h}} \right) \hfill \\ \end{gathered}$$

2.3 FG-O:

$$\begin{gathered} {\text{when: z}}\langle {\text{ r}}_{i} { + }\frac{h}{2} \hfill \\ {\text{A}}_{1} = 1 - \frac{{4D_{z} r_{i} }}{h},{\text{ A}}_{2} = \frac{{4D_{z} }}{h},{\text{ A}}_{3} = P_{m}^{2} \left( {\frac{{4D_{x} r_{i} }}{h} - 1} \right),{\text{ A}}_{4} = - 4P_{m}^{2} \frac{{D_{x} }}{h} \hfill \\ \frac{{k_{i} }}{{k_{m} }} = 1 + 4D_{i} \left( {\frac{{z - r_{i} }}{h}} \right) \hfill \\ {\text{when: z }}\rangle {\text{ r}}_{i} { + }\frac{h}{2} \hfill \\ {\text{A}}_{1} = 1 + \frac{{4D_{z} r_{o} }}{h},{\text{ A}}_{2} = - \frac{{4D_{z} }}{h},{\text{ A}}_{3} = - P_{m}^{2} \left( {\frac{{4D_{x} r_{o} }}{h} + 1} \right),{\text{ A}}_{4} = 4P_{m}^{2} \frac{{D_{x} }}{h} \hfill \\ \frac{{k_{i} }}{{k_{m} }} = 1 + 4D_{i} \left( {\frac{{r_{o} - z}}{h}} \right) \hfill \\ \end{gathered}$$

$$\begin{gathered} B_{2} = \frac{1}{{\int_{{r_{i} }}^{{r_{o} }} {\frac{{e^{{2\sqrt { - \, \frac{{A_{4} }}{{A_{2} }}} {\text{ z}}}} }}{{z\left( {A_{2} z + A_{1} } \right)H_{z}^{2} }}dz} }} \times \left\{ {\frac{{T_{o} }}{{H_{o} \times e^{{ - \sqrt { - \, \frac{{A_{4} }}{{A_{2} }}} \, r_{o} }} }} - \frac{{T_{i} }}{{H_{i} \times e^{{ - \sqrt { - \, \frac{{A_{4} }}{{A_{2} }}} {\text{ r}}_{i} }} }}} \right\} \hfill \\ C_{2} = \frac{{T_{i} }}{{HeunC\left( {\frac{{2A_{1} }}{{A_{2} }}\sqrt { - \frac{{A_{4} }}{{A_{2} }}} ,0,0,\frac{{A_{1} }}{{A_{2}^{3} }}\left( {A_{1} A_{4} - A_{2} A_{3} } \right),0, - \frac{{A_{2} }}{{A_{1} }}r_{i} } \right) \times e^{{ - \sqrt { - \, \frac{{A_{4} }}{{A_{2} }}} \, r_{i} }} }} \hfill \\ \end{gathered}$$

$$\begin{gathered} H_{z} = HeunC\left( {\frac{{2A_{1} }}{{A_{2} }}\sqrt { - \frac{{A_{4} }}{{A_{2} }}} ,0,0,\frac{{A_{1} }}{{A_{2}^{3} }}\left( {A_{1} A_{4} - A_{2} A_{3} } \right),0, - \frac{{A_{2} }}{{A_{1} }}z} \right) \hfill \\ H_{o} = HeunC\left( {\frac{{2A_{1} }}{{A_{2} }}\sqrt { - \frac{{A_{4} }}{{A_{2} }}} ,0,0,\frac{{A_{1} }}{{A_{2}^{3} }}\left( {A_{1} A_{4} - A_{2} A_{3} } \right),0, - \frac{{A_{2} }}{{A_{1} }}r_{o} } \right) \hfill \\ H_{i} = HeunC\left( {\frac{{2A_{1} }}{{A_{2} }}\sqrt { - \frac{{A_{4} }}{{A_{2} }}} ,0,0,\frac{{A_{1} }}{{A_{2}^{3} }}\left( {A_{1} A_{4} - A_{2} A_{3} } \right),0, - \frac{{A_{2} }}{{A_{1} }}r_{i} } \right) \hfill \\ \end{gathered}$$

$$\begin{gathered} B_{3} = \frac{1}{\Delta A}\left| {\begin{array}{*{20}c} {T_{2} } & {K_{o} \left( {m_{a} z} \right)} \\ {T_{ai} } & {K_{o} \left( {m_{a} \left( {r_{o} - h_{p} } \right)} \right)} \\ \end{array} } \right| \, C_{3} = \frac{1}{\Delta A}\left| {\begin{array}{*{20}c} {I_{o} \left( {m_{a} z} \right)} & {T_{2} } \\ {I_{o} \left( {m_{a} \left( {r_{o} - h_{p} } \right)} \right)} & {T_{ai} } \\ \end{array} } \right| \hfill \\ \Delta A = \left| {\begin{array}{*{20}c} {I_{o} \left( {m_{a} z} \right)} & {K_{o} \left( {m_{a} z} \right)} \\ {I_{o} \left( {m_{a} \left( {r_{o} - h_{p} } \right)} \right)} & {K_{o} \left( {m_{a} \left( {r_{o} - h_{p} } \right)} \right)} \\ \end{array} } \right| \hfill \\ Where{\text{ T}}_{ai} = \left. {T_{a} } \right|_{{z = r_{o} - h_{p} }} \hfill \\ \end{gathered}$$