Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams

An inhomogeneous beam model incorporating size effects is formulated based on the nonlocal strain gradient theory of elasticity within the framework of a third-order shear deformation beam theory to analyze the size dependency in vibration behavior of postbuckled laminated functionally graded (FG) micro-/nanobeams made from graphene platelet-reinforced composite (GPLRC). The graphene platelets are randomly dispersed in each individual layer in such a way that the weight fraction of the nanofiller varies layerwise on the basis of different patterns of FG dispersion. Based upon the Halpin-Tsai micromechanical scheme, the effective material properties of laminated FG-GPLRC micro-/nanobeams are extracted. With the aid of the Hamilton's principle, the nonlocal strain gradient equations of motion are constructed. After that, an improved perturbation technique is put to use to capture the small-scale effects on the fundamental frequencies of laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. The size-dependent natural frequencies of laminated third-order shear deformable FG-GPLRC micro-/nanobeams are obtained as function of applied axial compressive load within both the prebuckling and postbuckling domains. It is seen that in the prebuckling regime, the softening-stiffness effect of the nonlocality causes to reduce the natural frequencies of laminated FG-GPLRC micro-/nanobeam, but the hardening-stiffness influence of strain gradient size effect leads to increase the natural frequencies. However, by moving to the postbuckling regime, this pattern becomes vice versa, as the nonlocal size dependency leads to increase the natural frequencies of micro-/nanobeam while the strain gradient size effect causes to reduce them.