A coupled nonlinear nonlocal strain gradient theory for functionally graded Timoshenko nanobeams

A coupled nonlinear nonlocal strain gradient theory (NSGT) is developed for functionally graded Timoshenko nanoscale (FGTN) beams. This highly nonlinear-scale-dependent continuum model is solved numerically for dynamical responses for the first time. Hamilton’s energy minimisation technique is used for rotation-equation of the nanobeam in addition to axial-displacement and transverse-displacement equations; these equations are coupled both nonlinearly and linearly, where decoupling is not possible. The functional nature of the material of the nanobeam is incorporated via the Mori–Tanaka mixture scheme. Small-scale influences are taken into account via the NSGT which is able to incorporate both softening/hardening effects. The coupled/nonlinear model is reduced to a high-dimensional system using Galerkin’s method. The dynamics of the FGTN beam is analysed and the influence of functionally graded nature as well as the softening/hardening effects due to the NSGT is highlighted.