Buckling analysis of porous functionally graded GPL-reinforced conical shells subjected to combined forces

Abstract

This paper presents the analytical process of porous functionally graded graphene-reinforced truncated conical shells subjected to hydrostatic pressure and axial tension. Three types of graphene platelet (GPL) dispersion and three patterns of porous distribution were considered. A modified model, in which the volume fraction of the pores is regarded as the essential parameter, was built to evaluate the material properties. The static stability equations of the shells, coupled with the effect of the combined forces, were derived. The Galerkin integrate technique was employed to obtain the critical buckling hydrostatic pressure and axial tension. After the present method was validated, the influences of pores, GPLs, and shell geometrical characteristics were investigated in the parametric studies. The results show that the critical hydrostatic pressure and axial tension can be elevated by increasing the porous coefficient, semi-vertex, and length–thickness ratio. On the contrary, the critical pressure and tension are decreased with the rise of GPL mass fraction.

Appendix

$$L_{11} () = A_{11} x\frac{{\partial^{2} ()}}{{\partial x^{2} }} + A_{11} \frac{\partial ()}{{\partial x}} - A_{22} \frac{()}{x} + \frac{1}{{x\sin^{2} \alpha }}A_{66} \frac{{\partial^{2} ()}}{{\partial \theta^{2} }}$$

$$L_{12} () = \left( {A_{12} + A_{66} } \right)\frac{1}{\sin \alpha }\frac{{\partial^{2} ()}}{\partial x\partial \theta } - (A_{22} + A_{66} )\frac{1}{x\sin \alpha }\frac{\partial ()}{{\partial \theta }}$$

$$\begin{aligned} L_{13} () & = - B_{11} x\frac{{\partial^{3} ()}}{{\partial x^{3} }} - (B_{12} + 2B_{66} )\frac{1}{{x\sin^{2} \alpha }}\frac{{\partial^{3} ()}}{{\partial x\partial \theta^{3} }} - B_{11} \frac{{\partial^{2} ()}}{{\partial x^{2} }} \\ & \quad + \left( {B_{12} + B_{22} + 2B_{66} } \right)\frac{1}{{x^{2} \sin^{2} \alpha }}\frac{{\partial^{2} ()}}{{\partial \theta^{2} }} + A_{12} \cot \alpha \frac{\partial ()}{{\partial x}} \\ & \quad + B_{22} \frac{1}{x}\frac{\partial ()}{{\partial x}} - A_{22} \cot \alpha \frac{1}{x}() \\ \end{aligned}$$

$$L_{21} () = \left( {A_{12} + A_{66} } \right)\frac{1}{\sin \alpha }\frac{{\partial^{2} ()}}{\partial x\partial \theta } + (A_{22} + A_{66} )\frac{1}{x\sin \alpha }\frac{\partial ()}{{\partial \theta }}$$

$$L_{22} () = A_{22} \frac{1}{{x\sin^{2} \alpha }}\frac{{\partial^{2} ()}}{{\partial \theta^{2} }} + A_{66} x\frac{{\partial^{2} ()}}{{\partial x^{2} }} + A_{66} \frac{\partial ()}{{\partial x}} - \frac{1}{x}A_{66} ()$$

$$\begin{aligned} L_{23} () & = - \left( {B_{12} + 2B_{66} } \right)\frac{1}{\sin \alpha }\frac{{\partial^{3} ()}}{{\partial x^{2} \partial \theta }} - B_{22} \frac{1}{{x^{2} \sin^{2} \alpha }}\frac{{\partial^{3} ()}}{{\partial \theta^{3} }} \\ & \quad - B_{22} \frac{1}{x\sin \alpha }\frac{{\partial^{2} ()}}{\partial x\partial \theta } + A_{22} \cot \alpha \frac{1}{x\sin \alpha }\frac{\partial ()}{{\partial \theta }} \\ \end{aligned}$$

$$\begin{aligned} L_{31} () & = B_{11} x\frac{{\partial^{3} ()}}{{\partial x^{3} }} + (B_{12} + 2B_{66} )\frac{1}{{x\sin^{2} \alpha }}\frac{{\partial^{3} ()}}{{\partial x\partial \theta^{2} }} + 2B_{11} \frac{{\partial^{2} ()}}{{\partial x^{2} }} \\ & \quad + B_{22} \frac{1}{{x^{2} \sin^{2} \alpha }}\frac{{\partial^{2} ()}}{{\partial \theta^{2} }} - (A_{12} \cot \alpha + \frac{1}{x}B_{22} )\frac{\partial ()}{{\partial x}} + \frac{1}{{x^{2} }}B_{22} () - A_{22} \cot \alpha \frac{1}{x}() \\ \end{aligned}$$

$$\begin{aligned} L_{32} () & = \left( {B_{11} + 2B_{66} } \right)\frac{1}{\sin \alpha }\frac{{\partial^{3} ()}}{{\partial x^{2} \partial \theta }} + B_{22} \frac{1}{{x^{2} \sin^{3} \alpha }}\frac{{\partial^{3} ()}}{{\partial \theta^{3} }} \\ & \quad - B_{22} \frac{1}{x\sin \alpha }\frac{{\partial^{2} ()}}{\partial x\partial \theta } + \left( {B_{22} \frac{1}{{x^{2} \sin \alpha }} - A_{22} \cot \alpha \frac{1}{x\sin \alpha }} \right)\frac{\partial ()}{{\partial \theta }} \\ \end{aligned}$$

$$\begin{aligned} L_{33} () & = - D_{11} x\frac{{\partial^{4} ()}}{{\partial x^{4} }} - D_{22} \frac{1}{{x^{3} \sin^{4} \alpha }}\frac{{\partial^{4} ()}}{{\partial \theta^{4} }} - (D_{12} + 2D_{66} )\frac{2}{{x\sin^{2} \alpha }}\frac{{\partial^{4} ()}}{{\partial x^{2} \partial \theta^{2} }} \\ & \quad + \left( {D_{12} + 2D_{66} } \right)\frac{2}{{x^{2} \sin^{2} \alpha }}\frac{{\partial^{3} ()}}{{\partial x\partial \theta^{2} }} - 2D_{11} \frac{{\partial^{3} ()}}{{\partial x^{3} }} + \left( {\frac{1}{x}D_{22} + 2B_{12} \cot \alpha + xK_{2} } \right)\frac{{\partial^{2} ()}}{{\partial x^{2} }} \\ & \quad + \left[ {B_{22} {\text{cot}}\alpha \frac{2}{{x^{2} \sin^{2} \alpha }} - \left( {D_{12} + 2D_{66} + D_{22} } \right)\frac{2}{{x^{3} \sin^{2} \alpha }} + \frac{{K_{2} }}{x\sin \alpha }} \right]\frac{{\partial^{2} ()}}{{\partial \theta^{2} }} \\ & \quad K_{2} \frac{\partial ()}{{\partial x}} - D_{22} \frac{1}{{x^{2} }}\frac{\partial ()}{{\partial x}} + B_{22} \cot \alpha \frac{1}{{x^{2} }}() - A_{22} \cot^{2} \alpha \frac{1}{x}() - xK_{1} () \\ \end{aligned}$$

$$L_{34} () = - \frac{1}{2}\left( {x\frac{\partial ()}{{\partial x}} + x^{2} \frac{{\partial^{2} ()}}{{\partial x^{2} }}} \right)$$

$$L_{35} () = - \frac{2}{\sin 2\alpha }\frac{{\partial^{2} ()}}{{\partial \theta^{2} }}$$

$$L_{36} () = \frac{1}{2}s_{1}^{2} \tan \alpha \frac{{\partial^{2} ()}}{{\partial x^{2} }}$$

where the stiffness coefficients $A_{ij} ,B_{ij}$ and $D_{{{\text{ij}}}}$ can be calculated by

$$\left( {A_{11} ,B_{11} ,D_{11} } \right)^{T} = \left( {A_{22} ,B_{22} ,D_{22} } \right)^{T} = \int_{ - 0.5h}^{0.5h} {(1,z,z^{2} )\frac{E(z)}{{1 - \mu^{2} (z)}}{\text{d}}z}$$

$$\left( {A_{12} ,B_{12} ,D_{12} } \right)^{T} = \left( {A_{21} ,B_{21} ,D_{21} } \right)^{T} = \int_{ - 0.5h}^{0.5h} {(1,z,z^{2} )\frac{\mu (z)E(z)}{{1 - \mu^{2} (z)}}{\text{d}}z}$$

$$\left( {A_{66} ,B_{66} ,D_{66} } \right)^{T} = \int_{ - 0.5h}^{0.5h} {(1,z,z^{2} )\frac{E(z)}{{2[1 + \mu (z)]}}\,{\text{d}}z}$$