Higher-order finite strip method (H-FSM) with nonlocal strain gradient theory for analyzing bending and free vibration of orthotropic nanoplates

Abstract

This paper develops a size-dependent Kirchhoff plate model for bending and free vibration analyses using a semi-analytical higher-order finite strip method (H-FSM) based on the nonlocal strain gradient theory (NSGT). To satisfy the various longitudinal boundary conditions, the continuous trigonometric function series and the interpolation polynomial functions are employed in the transverse direction. In solving nanoplate problems using the H-FSM, the higher-order polynomial shape functions (higher-order Hermitian shape functions) are utilized to evaluate the second derivatives, in addition to the displacement and first derivative. The stiffness and mass matrices, and force vector of the nanoplates are derived using the weighted residual method. A numerical study is conducted to investigate the impact of different factors, such as boundary conditions, nonlocal and strain gradient parameters, aspect ratio, and types of transverse loading. The Navier solution is utilized to analyze the effects of material length scale parameters on bending and free vibration responses of nanoplates for preliminary comparisons. The numerical results show that, when the transverse load on the nanoplate is uniform or hydrostatic and the plate has a CCCC boundary condition, the nonlocal effect does not affect the deflection results and is the same as the obtained results in the local mode.