Post-buckling dynamics of Timoshenko microbeams under axial loads

Abstract

The nonlinear buckling and post-buckling dynamics of a Timoshenko microbeam under an axial load are examined in this paper. The microbeam is assumed to be subject to an axial force along with a distributed harmonically-varying transverse force. As the axial load is increased, in the absence of the transverse load, the stability of the microbeam is lost by a super-critical pitchfork bifurcation at the critical axial load, leading to a static divergence (buckling). The post-buckling state, due to a sufficiently high axial load, is obtained numerically; the resonant response over the buckled state, due to the harmonic transverse loading, is also examined. More specifically, Hamilton’s principle is employed to derive the equations of motion taking into account the effect of the length-scale parameter via the modified couple stress theory. These partial differential equations are then discretized by means of the Galerkin scheme, yielding a set of ordinary differential equations. The resultant equations are then solved via a continuation technique as well as a direct time-integration method. Results are shown through bifurcation diagrams, frequency-responses, and force-response curves. Points of interest in the parameter space in the form of time histories, phase-plane portraitists, and fast Fourier transforms are also highlighted. Finally, the effect of taking into account the length-scale parameter on the buckling and post-buckling behaviour of the system is highlighted by comparing the results obtained through use of the classical and modified couple stress theories.