Nonlinear dynamic stability analysis of three-dimensional graphene foam-reinforced polymeric composite cylindrical shells subjected to periodic axial loading

Abstract

This paper investigates the nonlinear dynamic stability of three-dimensional graphene foam (3D-GrF)-reinforced polymeric composite (RPC) cylindrical shells under the periodic axial loading. Three types of foam distribution are considered and the effective Young’s modulus, Poisson’s ratio and mass density of 3D-GrFRPC shells are obtained by using the mixing rule. Hamilton’s principle is implemented to derive the differential equations of motion on the basis of Donnell’s shell theory and von Kàrmàn geometric nonlinearity. In the framework of the Galerkin method and Airy stress function, the nonlinear transverse vibration differential equation is transformed to the Mathieu–Hill equation. Moreover, the explicit expressions of steady-state vibration amplitude of 3D-GrFRPC shells are obtained via Bolotin’s method. Finally, the effects of foam coefficient, foam distribution, dynamic load factor, static load factor and shell geometry parameters on the nonlinear dynamic stability of 3D-GrFRPC cylindrical shells are discussed.

Appendix

$$ M_{mn} = - 16A_{11}^{2} L^{4} \pi R^{4} \left[ {A_{11} A_{66} L^{4} n^{4} - \left( {A_{12}^{2} - A_{11}^{2} + 2A_{12} A_{66} } \right)L^{2} m^{2} n^{2} \pi^{2} R^{2} + A_{11} A_{66} m^{4} \pi^{4} R^{4} } \right]\rho_{t} $$

(A.1)

$$ Q_{mn} = 16A_{11}^{2} L^{2} m^{2} \pi^{3} R^{4} \left[ { - A_{11} A_{66} L^{4} n^{4} + \left( {A_{12}^{2} - A_{11}^{2} + 2A_{12} A_{66} } \right)L^{2} m^{2} n^{2} \pi^{2} R^{2} - A_{11} A_{66} m^{4} \pi^{4} R^{4} } \right] $$

(A.2)

$$ \eta_{mn} = \pi \left[ {A_{11} \left( {L^{4} n^{4} + m^{4} \pi^{4} R^{4} } \right)} \right]\left( {A_{12}^{2} - A_{11}^{2} } \right)\left[ {A_{11} A_{66} L^{4} n^{4} - \left( {A_{12}^{2} - A_{11}^{2} + 2A_{12} A_{66} } \right)L^{2} m^{2} n^{2} \pi^{2} R^{2} + A_{11} A_{66} m^{4} \pi^{4} R^{4} } \right] $$

(A.3)

$$ \begin{aligned} K_{mn} = & 16A_{11}^{2} \pi \left\{ {A_{66} \left( {B_{11}^{2} - A_{11} D_{11} } \right)L^{8} n^{8} } \right. + \left\{ {B_{11} \left[ {A_{11} B_{11} - 2A_{12} \left( {B_{12} + 2B_{66} } \right)} \right]} \right. + A_{12} \left( {A_{12} + 2A_{66} } \right)D_{11} \\ & \quad + \left. {A_{11} \left[ {\left( {B_{12} + 2B_{66} } \right)^{2} - A_{11} D_{11} - 2A_{66} \left( {D_{12} + 2D_{66} } \right)} \right]} \right\}L^{6} m^{2} n^{6} \pi^{2} R^{2} + 2A_{66} \left( { - A_{11} B_{12} + A_{12} B_{11} } \right)L^{6} m^{2} n^{4} \pi^{2} R^{3} \\ & \quad - \left\{ {2A_{66} B_{11}^{2} - 2A_{11} B_{12} B_{11} - 4A_{11} B_{11} B_{66} + A_{11} A_{66} D_{11} - 2A_{12}^{2} \left( {D_{12} + 2D_{66} } \right) + A_{11} \left[ { - 2B_{11} } \right.\left( {B_{12} + 2B_{66} } \right)} \right. \\ & \quad + \left. {A_{66} D_{11} + 2A_{11} D_{12} + 4A_{11} D_{66} } \right] + \left. {2A_{12} \left[ {B_{11}^{2} + \left( {B_{12} + 2B_{66} } \right)^{2} - 2A_{66} \left( {D_{12} + 2D_{66} } \right)} \right]} \right\}L^{4} m^{4} n^{4} \pi^{4} R^{4} \\ & \quad - 2\left( {2A_{11} A_{66} B_{11} - 2A_{12} A_{66} B_{12} + 2A_{12}^{2} B_{66} - 2A_{11}^{2} B_{66} } \right)L^{4} m^{4} n^{2} \pi^{4} R^{5} \\ & \quad + L^{2} m^{4} \pi^{4} \left\{ {\left( {A_{12}^{2} - A_{11}^{2} } \right)A_{66} L^{2} + \left\{ { - 2A_{12} B_{11} \left( {B_{12} + 2B_{66} } \right) + A_{12}^{2} D_{11} + 2A_{12} A_{66} D_{11} } \right.} \right. \\ & \quad + \left. {A_{11} \left( {B_{11}^{2} - A_{11} D_{11} } \right) + \left. {A_{11} \left[ {\left( {B_{12} + 2B_{66} } \right)^{2} - 2A_{66} \left( {D_{12} + 2D_{66} } \right)} \right]} \right\}m^{2} n^{2} \pi^{2} } \right\}R^{6} \\ & \quad + 2A_{66} \left( {A_{12} B_{11} - A_{11} B_{12} } \right)L^{2} m^{6} \pi^{6} R^{7} + \left. {A_{66} \left( {B_{11}^{2} - A_{11} D_{11} } \right)m^{8} \pi^{8} R^{8} } \right\}. \\ \end{aligned} $$

(A.4)