Three-dimensional nonlinear size-dependent behaviour of Timoshenko microbeams

Abstract

The geometrically nonlinear size-dependent behaviour of a Timoshenko microbeam is examined numerically, taking into account the coupled longitudinal–transverse displacements as well as the rotation. The strain energy of a Timoshenko microbeam is obtained based on the modified couples stress theory. Hamilton’s principle is then employed to derive the nonlinear partial differential equations of motion for the longitudinal, transverse, and rotational motions. The Galerkin scheme is applied to these nonlinear partial differential equations, resulting in a set of nonlinear ordinary differential equations with coupled terms. The nonlinear resonant response of the system is examined by solving the discretized equations of motion via the pseudo-arclength continuation technique and constructing the frequency–response and force–response curves. In particular, the effect of the length scale parameter is investigated by comparing the results obtained using the modified couple and classical theories. The frequency–response curves of the present model are compared to those of the one in which the longitudinal displacement is neglected so as to highlight the importance of taking into account the longitudinal displacement. The effect of other system parameters on the frequency–response and force–response curves is also investigated.