Nonlocal strain gradient nonlinear primary resonance of micro/nano-beams made of GPL reinforced FG porous nanocomposite materials

The current study deals with the size dependency in the nonlinear primary resonance of harmonic soft excited micro/nano-beams made of functionally graded (FG) porous nanocomposite materials reinforced with graphene platelets (GPLs). The size-dependent mathematical formulations are developed based upon the nonlocal strain gradient theory of elasticity within the framework of a refined hyperbolic shear deformation beam theory. On the basis of the closed-cell Gaussian-Random field scheme as well as the Halpin-Tsai micromechanical modeling, the mechanical properties of the FG porous nanocomposite material reinforced with GPLs are extracted corresponding to the uniform and three different FG patterns of porosity dispersion. The nonlocal strain gradient differential equations of motion including the geometrical nonlinearity are constructed using the variational approach. Thereafter, a numerical solution methodology based on 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 the FG porous nanocomposite micro/nano-beams reinforced with GPLs. It is found that the nonlocal size effect makes a reduction in the excitation amplitude associated with the both bifurcation points, especially for the first bifurcation point. However, the strain gradient size dependency causes to increase them. Also, it is displayed that by increasing the value of the porosity coefficient, the peak of the jump phenomenon associated with the soft harmonic excited FG porous nanocomposite micro/nano-beam is shifted to higher excitation frequency, which makes an enhancement in the hardening behavior related to the nonlinear primary resonance. Communicated by Eleonora Tubaldi.