Size-dependent nonlinear forced vibration of microbeam using two-phase local/nonlocal viscoelastic integral model with thermal effects

Combination strain-driven (εD) and stress-driven (σD) two-phase local/nonlocal integral models (TPNIM) with classic Kelvin-Voigt viscoelastic model, εD- and σD-two-phase local/nonlocal viscoelastic integral models (TPNVIM) are proposed to study the viscoelastic nonlinear forced vibration behaviors of microbeams in thermal environment. Based on von-Karman large deformation theory and the Hamilton’s principle, the differential governing equations of motion and boundary conditions are derived for nonlinear forced vibration of microbeam. Several non-dimensional variables are introduced to simplify the mathematical formulation. Nonlocal viscoelastic equations in integral form are transferred unitedly into equivalent differential form with constitutive constraints. The axial force of nonlocal microbeam with immovable ends is derived explicitly. Vibration frequency and corresponding linear vibration mode shape are derived through Laplace transformation for linear thermo-elastic vibration of microbeam using εD- and σD-TPNIMs. The Ritz-Galerkin approach is utilized to derive the reduced differential governing equation for nonlinear vibration of microbeam, which has exact same form as single-degree-of-freedom oscillator with cubic nonlinearity when material viscosity is neglected. The method of multiple scales (MMS) based on trigonometric functions is utilized to solve the reduced nonlinear differential equation approximately. The frequency-amplitude responses are addressed for primary and super-harmonic resonances of microbeam. The effects of viscous coefficient, amplitude of excitation force and environmental temperature variation as well as nonlocal length parameter on the nonlinear vibration behavior of microbeam are investigated numerically for different boundary conditions.