Nonlocal strain gradient forced vibrations of FG-GPLRC nanocomposite microbeams

In the current investigation, based upon the nonlocal strain gradient theory of elasticity, an inhomogeneous size-dependent beam model is formulated within the framework of a refined hyperbolic shear deformation beam theory. Thereafter, via the constructed nonlocal strain gradient refined beam model, the nonlinear primary resonance of laminated functionally graded graphene platelet-reinforced composite (FG-GPLRC) microbeams under external harmonic excitation is studied in the presence of the both hardening-stiffness and softening-stiffness size effects. The graphene platelets are randomly dispersed in each individual layer in such a way that the weight fraction of the reinforcement varies on the basis of different patterns of FG dispersion. Based upon the Halpin-Tsai micromechanical scheme, the effective material properties of laminated FG-GPLRC microbeams are achieved. By putting the Hamilton's principle to use, the nonlocal strain gradient equations of motion are developed. After that, a numerical solving process using the generalized differential quadrature (GDQ) method together with the Galerkin technique is employed to obtain the nonlocal strain gradient frequency response and amplitude response associated with the nonlinear primary resonance of laminated FG-GPLRC microbeams. It is found that the nonlocality size effect leads to an increase in the peak of the jump phenomenon and the associated excitation frequency, while the strain gradient size dependency results in a reduction in both of them.