Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects

•A model is derived to study the post-buckling of functionally graded nanobeams.•The model incorporates nonlocal stress and microstructure-dependent strain gradient effects.•Closed-form solutions for post-buckled configuration and critical buckling force are derived.•Stiffness-hardening or stiffness-softening effects can be found and depend on the values of small-scaled parameters.

On the basis of the nonlocal strain gradient theory, a size-dependent Euler-Bernoulli beam model is formulated and devoted to investigating the scaling effect on the post-buckling behaviors of functionally graded (FG) nanobeams with the von Kármán geometric nonlinearity. The developed beam model can incorporate the scaling effect of both nonlocal long-range force and microstructure-dependent strain mechanism. To simplify the redundancy of the governing equation and derive the closed-form solutions, a physical neutral surface is applied for removing the bending-stretching coupling due to geometric nonlinearity and the coupling rigidity between the extensional and bending rigidities of the though-thickness FG material. The closed-form solutions for the post-buckled configuration and the critical buckling force (CBF) are deduced in the case of hinged-hinged boundary conditions. The effects of scaling parameters and material property variation on the post-buckled configuration and the CBF are investigated in detail. It is found that the stiffness-hardening or stiffness-softening effect is dependent of the values of scaling parameters.