Nonlocal temperature-dependent postbuckling behavior of FG-CNT reinforced nanoshells under hydrostatic pressure combined with heat conduction

Abstract

The objective of this paper is to anticipate the size-dependent nonlinear instability response of functionally graded carbon nanotube (FG-CNT) reinforced nanocomposite nanoshells with temperature-dependent material properties under combination of hydrostatic pressure and through-thickness heat conduction. To this purpose, a new size-dependent shell model is constructed based upon implementation of the Eringen’s nonlocal continuum elasticity theory in conjunction with von Karman kinematics nonlinearity into a refined shell theory with exponential function to distribute the shear deformation. In addition to the uniform distribution (UD) of CNT reinforcements, three FG patterns are also considered namely FG-A, FG-V and FG-X. Moreover, the temperature variation due to the through-thickness heat conduction is evaluated using polynomial series. Subsequently, by transforming the problem to a boundary layer-type one and with the aid of a two-stepped perturbation solving process, explicit expressions for nonlocal temperature-dependent load–deflection and load-shortening curves of exponential shear deformable FG-CNT reinforced nanoshells with and without initial geometric imperfection are presented. It is indicated that the reduction in the value of critical buckling pressure due to the nonlocality size effect is more prominent for FG-A distribution pattern in comparison with other ones, especially for thinner FG-CNT reinforced nanoshells.

Appendices

Appendix A

$$ \begin{aligned} \varphi_{1} = \frac{{A_{11}^{*} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }}, \varphi_{2} = \frac{{A_{12}^{*} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }}, \varphi_{3} = \frac{{A_{11}^{*} B_{11}^{*} - A_{12}^{*} B_{12}^{*} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }} \hfill \\ \varphi_{4} = \frac{{A_{11}^{*} B_{12}^{*} - A_{12}^{*} B_{11}^{*} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }}, \varphi_{5} = \frac{{A_{11}^{*} B_{11}^{**} - A_{12}^{*} B_{12}^{**} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }}, \varphi_{6} = \frac{{A_{11}^{*} B_{12}^{**} - A_{12}^{*} B_{11}^{**} }}{{\left( {A_{11}^{*} } \right)^{2} - \left( {A_{12}^{*} } \right)^{2} }} \hfill \\ \varphi_{7} = \frac{1}{{A_{66}^{*} }} , \varphi_{8} = \frac{{B_{66}^{*} }}{{A_{66}^{*} }} , \varphi_{9} = \frac{{B_{66}^{**} }}{{A_{66}^{*} }} \hfill \\ \varphi_{10} = D_{11}^{*} - B_{11}^{*} \varphi_{3} - B_{12}^{*} \varphi_{4} , \varphi_{11} = D_{12}^{*} - B_{11}^{*} \varphi_{3} - B_{12}^{*} \varphi_{4} \hfill \\ \varphi_{12} = D_{66}^{*} - B_{66}^{*} \varphi_{8} , \varphi_{13} = B_{11}^{*} \varphi_{5} + B_{12}^{*} \varphi_{6} - D_{11}^{**} \hfill \\ \varphi_{14} = B_{12}^{*} \varphi_{5} + B_{11}^{*} \varphi_{6} - D_{12}^{*} , \varphi_{15} = B_{66}^{**} \varphi_{9} - D_{66}^{**} \hfill \\ \varphi_{16} = B_{11}^{**} \varphi_{3} + B_{12}^{**} \varphi_{4} - D_{11}^{**} , \varphi_{17} = B_{12}^{**} \varphi_{3} + B_{11}^{**} \varphi_{4} - D_{12}^{**} \hfill \\ \varphi_{18} = B_{66}^{**} \varphi_{8} - D_{66}^{** } , \varphi_{19} = G_{66}^{*} - B_{66}^{**} \varphi_{9} \hfill \\ \end{aligned} $$

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This point should be noted that the parameters of $\vartheta_{i} (i = 1, \ldots ,19)$ are the dimensionless form of $\varphi_{i}$.

The solutions in asymptotic forms for each of independent variables are extracted as below

$$ \begin{aligned} W = {\mathcal{A}}_{00}^{\left( 0 \right)} + \varepsilon^{3/2} \left[ {{\mathcal{A}}_{00}^{(3/2)} - {\mathcal{A}}_{00}^{(3/2)} \left( {\sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + { \cos }\left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} - {\mathcal{A}}_{00}^{(3/2)} \left( {\sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + { \cos }\left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right] + \varepsilon^{2} \left[ {{\mathcal{A}}_{00}^{\left( 2 \right)} + {\mathcal{A}}_{11}^{\left( 2 \right)} \sin \left( {mX} \right)\sin \left( {nY} \right) - {\mathcal{A}}_{00}^{(2)} \left( {\sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + { \cos }\left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} - {\mathcal{A}}_{00}^{(2)} \left( {\sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + { \cos }\left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right] \hfill \\ + \varepsilon^{3} \left[ {{\mathcal{A}}_{00}^{\left( 3 \right)} + {\mathcal{A}}_{11}^{\left( 3 \right)} \sin \left( {mX} \right)\sin \left( {nY} \right)} \right] \hfill \\ + \varepsilon^{4} \left[ {{\mathcal{A}}_{00}^{\left( 4 \right)} + {\mathcal{A}}_{11}^{\left( 4 \right)} \sin \left( {mX} \right)\sin \left( {nY} \right) + {\mathcal{A}}_{20}^{\left( 4 \right)} \cos \left( {2mX} \right) + {\mathcal{A}}_{02}^{(4)} { \cos }(2nY)} \right] + O\left( {\varepsilon^{5} } \right) \hfill \\ \end{aligned} $$

(41)

$$ \begin{aligned} F = - {\mathcal{B}}_{00}^{(0)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right) + \epsilon \left[ { - {\mathcal{B}}_{00}^{(1)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right)} \right] \hfill \\ + \varepsilon^{2} \left[ { - {\mathcal{B}}_{00}^{(2)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right) + {\mathcal{B}}_{11}^{(2)} \sin \left( {mX} \right)\sin \left( {nY} \right)} \right] \hfill \\ + \varepsilon^{5/2} \left[ {{\mathcal{A}}_{00}^{(3/2)} \left( {{\fancyscript{b}}_{10}^{(2)} \sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + {\fancyscript{b}}_{01}^{(2)} { \cos }\left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} + {\mathcal{A}}_{00}^{(3/2)} \left( {{\fancyscript{b}}_{10}^{(2)} \sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + {\fancyscript{b}}_{01}^{(2)} { \cos }\left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right] \hfill \\ + \varepsilon^{3} \left[ { - {\mathcal{B}}_{00}^{(3)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right) + {\mathcal{A}}_{00}^{(2)} \left( {{\fancyscript{b}}_{10}^{(3)} \sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + {\fancyscript{b}}_{01}^{(3)} { \cos }\left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} + {\mathcal{A}}_{00}^{(2)} \left( {{\fancyscript{b}}_{10}^{(3)} \sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + {\fancyscript{b}}_{01}^{(3)} { \cos }\left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right] \hfill \\ + \varepsilon^{4} \left[ { - {\mathcal{B}}_{00}^{\left( 4 \right)} \left( {\beta^{2} X^{2} + \frac{{Y^{2} }}{2}} \right) + {\mathcal{B}}_{20}^{(4)} \cos \left( {2mX} \right) + {\mathcal{B}}_{02}^{(4)} \cos \left( {2nY} \right)} \right] + O\left( {\varepsilon^{5} } \right) \hfill \\ \end{aligned} $$

(42)

$$ \begin{aligned} \varPsi_{X} = \varepsilon^{2} \left[ {{\mathcal{C}}_{11}^{\left( 2 \right)} \cos \left( {mX} \right){ \sin }\left( {nY} \right) + \left( {c_{10}^{(2)} \sin \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right) + c_{01}^{(2)} \cos \left( {\frac{{\varGamma_{1} X}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} X}}{\sqrt \epsilon }}} + \left( {c_{10}^{(2)} \sin \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right) + c_{01}^{(2)} \cos \left( {\frac{{\varGamma_{1} \left( {\pi - X} \right)}}{\sqrt \epsilon }} \right)} \right)e^{{ - \frac{{\varGamma_{2} \left( {\pi - X} \right)}}{\sqrt \epsilon }}} } \right] \hfill \\ + \varepsilon^{3} \left[ {{\mathcal{C}}_{11}^{\left( 3 \right)} \cos \left( {mX} \right){ \sin }\left( {nY} \right)} \right] + \varepsilon^{4} \left[ {{\mathcal{C}}_{11}^{\left( 4 \right)} \cos \left( {mX} \right){ \sin }\left( {nY} \right) + {\mathcal{C}}_{20}^{\left( 4 \right)} { \sin }\left( {2mX} \right)} \right] + O\left( {\varepsilon^{5} } \right) \hfill \\ \end{aligned} $$

(43)

$$ \varPsi_{Y} = \varepsilon^{2} \left[ {{\mathcal{D}}_{11}^{(2)} \sin \left( {mX} \right){ \cos }(nY)} \right] + \varepsilon^{3} \left[ {{\mathcal{D}}_{11}^{\left( 3 \right)} \sin \left( {mX} \right){ \cos }\left( {nY} \right)} \right] + \varepsilon^{4} \left[ {{\mathcal{D}}_{11}^{\left( 4 \right)} \sin \left( {mX} \right){ \cos }\left( {nY} \right) + {\mathcal{D}}_{02}^{\left( 4 \right)} \sin \left( {2nY} \right)} \right] + O\left( {\varepsilon^{5} } \right) $$

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in which

$$ \varGamma_{1} = \sqrt {\frac{{\sqrt {\frac{1}{{\vartheta_{1} \vartheta_{10} + \vartheta_{4}^{2} }}} + \frac{{\vartheta_{4} }}{{\vartheta_{1} \vartheta_{10} + \vartheta_{4}^{2} }}}}{2}} , \varGamma_{2} = \sqrt {\frac{{\sqrt {\frac{1}{{\vartheta_{1} \vartheta_{10} + \vartheta_{4}^{2} }}} - \frac{{\vartheta_{4} }}{{\vartheta_{1} \vartheta_{10} + \vartheta_{4}^{2} }}}}{2}} $$

(45)

Appendix B

$$ {\mathcal{P}}_{q}^{(0)} = {\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{8} + {\mathcal{U}}_{2} {\mathcal{U}}_{8} \varepsilon^{2} $$

(46)

$$ \begin{aligned} {\mathcal{P}}_{q}^{(2)} = 8{\mathcal{U}}_{1} {\mathcal{U}}_{3} {\mathcal{U}}_{7} {\mathcal{U}}_{8} + \frac{{8{\mathcal{U}}_{1} {\mathcal{U}}_{3} {\mathcal{U}}_{8} \left( {{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{6} {\mathcal{U}}_{8} H_{20} + {\mathcal{U}}_{0} {\mathcal{U}}_{3} {\mathcal{U}}_{5} H_{20} } \right)}}{{{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{8} H_{20} - {\mathcal{U}}_{5} }} + \frac{{8{\mathcal{U}}_{1} {\mathcal{U}}_{3} \left( {{\mathcal{U}}_{0} {\mathcal{U}}_{6} + {\mathcal{U}}_{0}^{2} {\mathcal{U}}_{3} H_{20} } \right)}}{{{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{8} H_{20} - {\mathcal{U}}_{5} }} \hfill \\ + \frac{{8{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{3} ({\mathcal{U}}_{6} + {\mathcal{U}}_{0} {\mathcal{U}}_{3} )}}{{{\mathcal{U}}_{0} {\mathcal{U}}_{1} {\mathcal{U}}_{8} - {\mathcal{U}}_{5} }} + 16{\mathcal{U}}_{0} {\mathcal{U}}_{3} {\mathcal{U}}_{4} {\mathcal{U}}_{8} \hfill \\ \end{aligned} $$

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$$ \delta_{q}^{(0)} = \left[ {\frac{{\vartheta_{1} }}{2} - \vartheta_{2} + \left( {\frac{{\left( {2\vartheta_{1} \vartheta_{2} - \vartheta_{2}^{2} } \right)\varGamma_{2} }}{{\pi \vartheta_{1} \left( {\varGamma_{1}^{2} + \varGamma_{2}^{2} } \right)}}} \right)\varepsilon^{1/2} } \right]{\mathcal{P}}_{q} + \left[ {\left( {\frac{{3^{1/4} \left( {\varGamma_{1}^{2} + \varGamma_{2}^{2} } \right)\left( {2\vartheta_{1} - \vartheta_{2} } \right)^{2} }}{{6\pi \varGamma_{2} }}} \right)\epsilon } \right]{\mathcal{P}}_{q}^{2} $$

(48)

$$ \delta_{q}^{(2)} = \left[ {\frac{{3^{3/4} m^{2} }}{32}} \right]\varepsilon^{ - 3/2} $$

(49)

$$ \delta_{q}^{T} = \left( {\frac{{3^{3/4} }}{4}\frac{{\alpha_{11} R\Delta T}}{h}} \right)\varepsilon^{1/2} $$

(50)

where

$$ H_{11} = 1 + \pi^{2} {\mathcal{G}}^{2} \left( {m^{2} + \beta^{2} n^{2} } \right) , H_{20} = 1 + 4\pi^{2} {\mathcal{G}}^{2} m^{2} $$

(51)

where ${\mathcal{U}}_{i} (i = 0, \ldots ,8)$ are constant parameters extracted via the perturbation sets of equations.

$$ {\mathcal{S}}_{1} = - \left[ {\left( {2\vartheta_{1} - \vartheta_{2} } \right)\left( {{\mathcal{P}}_{q}^{\left( 2 \right)} } \right)} \right] $$

(52)

$$ {\mathcal{S}}_{2} = - \left( {2\vartheta_{1} - \vartheta_{2} } \right)\left( {{\mathcal{P}}_{q}^{\left( 0 \right)} } \right) + \left( {\frac{{\alpha_{22} R\Delta T}}{h}} \right)\epsilon $$

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