Thermal and thermomechanical buckling of shear deformable FG-CNTRC cylindrical shells and toroidal shell segments with tangentially restrained edges

Abstract

This paper presents a simple and effective analytical approach to investigate buckling behavior of carbon nanotube-reinforced composite (CNTRC) cylindrical shells and toroidal shell segments surrounded by elastic media and subjected to elevated temperature, lateral pressure and thermomechanical load. The properties of constituents are assumed to be temperature-dependent, and effective properties of CNTRC are estimated according to extended rule of mixture. Carbon nanotubes (CNTs) are reinforced into matrix material such in a way that their volume fraction is varied in the thickness direction according to functional rules. Formulations are established within the framework of first-order shear deformation theory taking surrounding elastic media and tangential elasticity of edges into consideration. The solutions of deflection and stress function are assumed to satisfy simply supported boundary conditions, and Galerkin method is used to derive expressions of buckling loads. In thermal buckling analysis, an iteration algorithm is employed to evaluate critical temperatures. The effects of CNT volume fraction and distribution patterns, degree of in-plane edge constraints, geometrical parameters, preexisting loads and surrounding elastic foundations on the critical loads of nanocomposite shells are analyzed through a variety of numerical examples.

Appendices

Appendix A

The details of the coefficients $A_{3}^{*},B_{1}^{*}$ and $B_{2}^{*}$ in Eq. (22) are the following

$$\begin{aligned} A_{3}^{*}= & {} \frac{b_{24} }{b_{14} }\left( {\frac{\delta _{n}^{2} }{a}+\frac{\beta _{\mathrm{m}}^{2} }{R}} \right) -\frac{b_{34} }{b_{14}}, \nonumber \\ B_{1}^{*}= & {} \frac{b_{13} b_{32} -b_{12} b_{33} }{b_{22} b_{33} -b_{23} b_{32} }\left[ {\frac{b_{24} }{b_{14} }\left( {\frac{\delta _{n}^{2} }{a}+\frac{\beta _{\mathrm{m}}^{2} }{R}} \right) -\frac{b_{34} }{b_{14} }} \right] +\frac{b_{32} e_{51} \delta _{n} -b_{33} e_{41} \beta _{\mathrm{m}} }{b_{22} b_{33} -b_{23} b_{32} }K_{\mathrm{S}},\nonumber \\ B_{2}^{*}= & {} \frac{b_{12} b_{23} -b_{13} b_{22} }{b_{22} b_{33} -b_{23} b_{32} }\left[ {\frac{b_{24} }{b_{14} }\left( {\frac{\delta _{n}^{2} }{a}+\frac{\beta _{\mathrm{m}}^{2} }{R}} \right) -\frac{b_{34} }{b_{14} }} \right] +\frac{b_{23} e_{41} \beta _{\mathrm{m}} -b_{22} e_{51} \delta _{n} }{b_{22} b_{33} -b_{23} b_{32} }K_{\mathrm{S}}, \end{aligned}$$

(A1)

in which $b_{ij} \,(i=1\div 3,j=2\div 4)$ have their forms as those given in the work [30].

The related quantities in Eqs. (25) and (26) are given as follows

$$\begin{aligned} a_{13}= & {} \left( {\frac{n^{2}}{R_{h}^{3} }R_{a} +\frac{m^{2}\pi ^{2}}{R_{h}^{3} L_{R}^{2} }} \right) \bar{{A}}_{3}^{*} -\frac{m^{2}n^{2}}{R_{h}^{4} L_{R}^{2} }\bar{{a}}_{51} \pi ^{2}\bar{{A}}_{3}^{*} -\frac{m^{3}\pi ^{3}}{R_{h}^{3} L_{R}^{3} }\bar{{a}}_{11} \bar{{B}}_{1}^{*} -\frac{mn^{2}\pi }{R_{h}^{3} L_{R} }\bar{{a}}_{21} \bar{{B}}_{1}^{*}\nonumber \\&-\frac{m^{2}n\pi ^{2}}{R_{h}^{3} L_{R}^{2} }\bar{{a}}_{31} \bar{{B}}_{2}^{*} -\frac{n^{3}}{R_{h}^{3} }\bar{{a}}_{41} \bar{{B}}_{2}^{*} +K_{1} \frac{E_{0\mathrm{m}} }{R_{h}^{4} }+K_{2} \frac{E_{0\mathrm{m}} }{R_{h}^{2} }\left( {\frac{m^{2}\pi ^{2}}{R_{h}^{2} L_{R}^{2} }+\frac{n^{2}}{R_{h}^{2} }} \right) ,\nonumber \\ L_{R}= & {} L/R, \quad \left( {\bar{{a}}_{11},\bar{{a}}_{21} ,\bar{{a}}_{31},\bar{{a}}_{41} } \right) =\frac{1}{h^{3}}\left( {a_{11},a_{21},a_{31},a_{41} } \right) , \quad \bar{{a}}_{51} =\frac{a_{51} }{h},\nonumber \\ \bar{{A}}_{3}^{*}= & {} \frac{A_{3}^{*} }{h^{2}}\,, \quad \left( {\bar{{B}}_{1}^{*},\bar{{B}}_{2}^{*} } \right) =\left( {B_{1}^{*},B_{2}^{*} } \right) h, \quad \left( {K_{1},K_{2} } \right) =\frac{R^{2}}{E_{0\mathrm{m}} h^{3}}\left( {k_{1} R^{2},k_{2} } \right) , \end{aligned}$$

(A2)

in which $E_{0\mathrm{m}}$ is the value of $E^{\mathrm{m}}$ calculated at room temperature $T_{0} =300\,\hbox {K}$.

Appendix B

The coefficients $g_{11}$ and $g_{21}$ in Eq. (27) are evaluated as follows

$$\begin{aligned} g_{11} =\frac{R_{h}^{4} }{g_{31} }, \, g_{21} =-\frac{R_{h}^{3} }{\bar{{e}}_{11} g_{31} }\left[ {\bar{{e}}_{11} \bar{{e}}_{21T} -\nu _{12} \bar{{e}}_{21} \bar{{e}}_{11T} +\lambda \bar{{e}}_{11T} \left( {\nu _{12} \bar{{e}}_{21} +R_{a} \bar{{e}}_{11} } \right) } \right] , \end{aligned}$$

(B1)

in which

$$\begin{aligned} g_{31} =K_{1} E_{0\mathrm{m}} +R_{h}^{2} \bar{{e}}_{21} \left( {1-\nu _{12} \nu _{21} } \right) +\lambda R_{h}^{2} \left( {\nu _{12} \bar{{e}}_{21} +R_{a} \bar{{e}}_{11} } \right) \left( {\nu _{21} +R_{a} } \right) . \end{aligned}$$

(B2)

The detailed expressions for coefficients $g_{12},g_{22}$ and $g_{32}$ in Eq. (28) are as follows

$$\begin{aligned} g_{12}= & {} a_{13}, \quad g_{22} =\frac{\bar{{e}}_{21} }{R_{h}^{3} }n^{2}\left( {1-\nu _{12} \nu _{21} } \right) +\lambda \frac{\bar{{e}}_{11} }{R_{h} }\left( {\nu _{21} +R_{a} } \right) \left( {\frac{m^{2}\pi ^{2}}{R_{h}^{2} L_{R}^{2} }+\nu _{12} \frac{n^{2}\bar{{e}}_{21} }{R_{h}^{2} \bar{{e}}_{11} }} \right) , \nonumber \\ g_{32}= & {} \frac{n^{2}}{R_{h}^{2} \bar{{e}}_{11} }\left( {\bar{{e}}_{11} \bar{{e}}_{21T} -\nu _{12} \bar{{e}}_{21} \bar{{e}}_{11T} } \right) +\lambda \bar{{e}}_{11T} \left( {\frac{m^{2}\pi ^{2}}{R_{h}^{2} L_{R}^{2} }+\nu _{12} \frac{n^{2}\bar{{e}}_{21} }{R_{h}^{2} \bar{{e}}_{11} }} \right) . \end{aligned}$$

(B3)