Motion characteristics of bilayered extensible Timoshenko microbeams

The nonlinear motion characteristics of a bilayered Timoshenko microbeam is analysed taking into account all the translational (i.e. longitudinal and transverse) and rotational motions; the effect of size is included through use of the modified couple stress theory. Considering a continuous variation through the thickness for the displacement field, the geometrically nonlinear strain terms are developed. Relating these strain terms to stress terms, via use of constitutive relations, leads to a stress tensor in terms of the displacement field; the same is applied to the symmetric curvature tensor and the deviatoric part of the symmetric couple stress tensor by means of a constitutive relation incorporating size effects. The potential energy of the bilayered microbeam is formulated via the modified couple stress theory. The kinetic energy along with the work of an external dynamic force is formulated in terms of the displacement field of the bilayered microbeam. A dynamic equilibrium state is obtained via balancing the energy and work. Three-dimensional reduced-order models for the transverse, longitudinal, and rotational motions are obtained via Galerkin's method. These coupled models are solved via a continuation method coupled with direct time-integration. The resonant responses are constructed with special consideration to the effects of the length-scale parameter and the material percentage and thickness of each layer; a comparison is also made between the motion characteristics of the bilayered and monolayered microbeam.