Size-dependent nonlocal strain gradient modeling of hexagonal beryllium crystal nanoplates

In this paper, a size dependent numerical model for free vibration and bending analyses of hexagonal beryllium crystal (HBC) nanoplates is presented. Based on the nonlocal strain gradient theory (NSGT), both nonlocal and strain gradient effects are considered in the present model. In addition, the behavior of HBC material can be considered as an anisotropic one. By employing principle of virtual displacement (PVD), the governing equations of motion for a Galerkin weak for the higher-order shear deformation theory (HSDT) are deduced. Thereafter, the natural frequency and deflection of the HBC nanoplates are determined by solving those governing equations using an isogeometric analysis (IGA). Numerical results show that bending and vibration behaviors of the anisotropic nanoplates are affected by the geometry, boundary conditions, length-to-thickness ratios, exponential factor, nonlocal parameter and strain gradient parameter. On the other hand, when the material length scale parameter is smaller than or equal to the nonlocal parameter then the natural frequency predicted by nonlocal strain gradient theory is lower than that evaluated by classical continuum theory. Conversely, obtained results are higher than the referenced ones by classical continuum theory when the material length scale parameter is larger than or equal to the nonlocal parameter. Whilst, a contrary case occurs for the displacements. Furthermore, by respectively ignoring the strain gradient and nonlocal parameters, the pure nonlocal and strain gradient models can be recovered from the present model.