Theoretical thermal damping vibration analysis of functionally graded viscoelastic Timoshenko microbeam with integral nonlocal strain gradient model

Abstract

In this work, nonlocal strain gradient integral model is applied to investigate the free damping vibration analysis of functionally graded (FG) viscoelastic Timoshenko microbeam with immovable boundary conditions in thermal environment. The microbeam is assumed to be viscoelastic and modeled by Kelvin–Voigt model. The differential governing equations and corresponding boundary conditions are derived with the Hamilton’s principle. Combining nonlocal strain gradient integral model and Kelvin–Voigt viscoelastic model, the integral constitutive equations of nonlocal stress with thermal effect are derived and then converted into differential form plus constitutive constraints. The size-dependent axial force due to thermal expansion is explicitly derived through the immovable boundary conditions. The bending deflection and moment, cross-sectional rotation as well as shear force are explicitly derived with Laplace transformation for linear thermo-elastic vibration. Taking into account the boundary conditions as well as constitutive constraints, one gets a nonlinear equation with complex coefficients, from which one can determine the complex characteristic frequency. A two-step numerical method is proposed to solve the elastic vibration frequency and damping ratio. The influence of length scale parameters, viscos coefficient, FG index, thermal effect, vibration order and ratio between beam length and thickness on the vibration frequencies and damping ratio is investigated numerically for FG viscoelastic Timoshenko microbeams under different immovable boundary conditions.