Dynamics of functionally graded viscoelastic microbeams

Abstract

In this paper, a size-dependent continuum-based model is presented for the coupled nonlinear dynamics of extensible functionally graded (FG) microbeams with viscoelastic properties. The length-scale effect is incorporated based on the modified couple stress theory (MCST). Moreover, employing the Kelvin–Voigt viscoelastic model, viscous components are taken into consideration in both stress and deviatoric segments of the symmetric couple stress tensors. The variation of the material properties of the FG viscoelastic microbeam along the thickness is approximated with the use of the Mori–Tanaka homogenisation method. Both the transverse and longitudinal motion as well as inertial terms are included in the size-dependent continuum model and numerical calculations. The elastic potential energy, kinetic energy and the viscos work are obtained with the consideration of size effects. Using von Karman's strain-displacement relations together with Hamilton's principle, the coupled differential equations of motion are derived. Then, Galerkin's approach and a continuation technique are used in order to obtain the fundamental frequency and dynamic response of the FG viscoelastic microbeam. The effects of parameters such as the gradient index, excitation frequency, the amplitude of the harmonic load and viscoelastic parameters on the nonlinear frequency- and force-responses of the FG viscoelastic microbeams are investigated in details.