Viscoelastically coupled size-dependent behaviour of imperfect extensible microbeams

This paper aims at investigating the size-dependent nonlinear behaviour of a viscoelastic imperfect extensible microbeam taking into account both transverse and longitudinal displacements and inertia. The size-dependent potential energy is formulated in the framework of the modified couple stress theory. The works due to the viscous parts of the stress tensor and the deviatoric part of the couple stress tensor are obtained in terms of system parameters. The kinetic energy as well as the work of external dynamic loading is obtained as functions of the displacement field. Hamilton's principle is employed in order to balance the work and energy terms which results in the coupled nonlinear equations of motion for the longitudinal and transverse directions. A high-dimensional weighted-residual technique is employed so as to discretise the coupled equations of longitudinal and transverse motions and hence the continuous system with infinite number of degrees of freedom is truncated into a reduced-order model with sufficient degrees of freedom for accurate results capable of capturing almost all modal interactions. This high-dimensional nonlinear coupled reduced-order model is solved for the fundamental coupled nonlinear resonant response via use of a continuation method as well as direct time integration for response characterisation with special consideration to the coupled effect of the viscousity, initial imperfection, and length-scale parameter on the system response in the longitudinal and transverse directions. It is shown that the deviation between the response of viscoelastic and elastic systems is substantial for fairly large excitation forces.