Functionally graded microbeams: simultaneous presence of imperfection and viscoelasticity

Abstract

As the first endeavour, the coupled nonlinear mechanical behaviour of extensible functionally graded microbeams, when both viscoelasticity and imperfection are present, is investigated. The imperfect viscoelastic microbeam is subject to a transverse harmonic excitation load of a constant amplitude. The Kelvin–Voigt viscoelastic model and Mori–Tanaka homogenisation method are used together in order to describe the internal energy loss and the variation of the material properties of the microsystem along the transverse direction, respectively. The geometric imperfection is modelled by imposing an initial curvature in the transverse deformation of the viscoelastic microscale beam. Using the Euler–Bernoulli strain–displacement relations, the geometric nonlinearity is taken into account. The non-classical nonlinear equations of motion are derived on the basis of Hamilton's principle and the modified couple stress theory. The resulting equations are found to be coupled between transverse and longitudinal oscillations. Galerkin's technique and the method of pseudo-arclength continuation as well as direct time-integration approach are finally employed to solve the governing differential equations for oscillation frequencies and nonlinear dynamic response. It is found that the nonlinear forced oscillations of extensible functionally graded microbeams are greatly affected by the internal energy loss together with geometric imperfection. It is shown that the simultaneous presence of viscoelasticity and geometric imperfections governs both the amplitude and softness/hardness of the dynamical behaviour.