功能梯度多层石墨烯片增强复合材料梁的非线性自由振动

We aimed to study the nonlinear free vibration of a functionally graded multilayer graphene platelet-reinforced composite (GPLRC) beam on a nonlinear elastic foundation under the combined effect of axial forces and temperature field. Based on the principle of virtual displacements, the nonlinear governing equations of beams are established by using the first-order shear deformation beam theory and von Krmn type nonlinear strain-displacement relationship. The three equations are simplified into two nonlinear equations about deflection and rotation, which are discretized by using the two-step perturbation method to obtain the perturbation equations of each order, and the asymptotic solutions are derived step by step to obtain the expressions of the frequencies under the simply supported boundary conditions at both ends. Through the parameter analysis, the effects of the graphene platelet (GPLs) distribution pattern, geometry and size, weight fraction, elastic foundation stiffness parameter, temperature and axial force on the nonlinear free vibration are explored. The calculation results show that the higher GPL weight fraction, then the more obvious the enhancement effect. Using larger surface area of GPL and less single layer GPL, adding more GPL near at the top and the bottom, and increasing the linear elastic foundation stiffness parameter can improve the natural frequency better, thus reducing nonlinear frequency-to-linear frequency ratio. The ratio of nonlinear frequency to linear frequency increases with the increase of nonlinear elastic foundation stiffness, temperature, and axial compression force.