Free damping vibration of functionally graded porous viscoelastic nonlocal microbeam with thermal effect

Combining classic Kelvin–Voigt viscoelastic model and two-phase local/nonlocal integral models (TPNIM), two-phase local/nonlocal integral viscoelastic models (TPNIVM) are proposed to study the free thermo-damping vibration of functionally graded porous microbeam with four different porous distribution patterns. The differential governing equations, boundary conditions, and constitutive constraints are formulated. The axial stress due to environmental temperature variation is derived explicitly. Several nominal variables are introduced to simplify the mathematical formulae. Laplace transformation is applied to obtain the explicit expression for bending deflection and moment. On considering the boundary conditions and constitutive constraints, a complex coefficients nonlinear equation is obtained to determine the complex characteristic frequency. A two-step numerical method is proposed to solve the elastic vibration frequency and damping ratio. The influence of nonlocal parameters, porous distribution parameters, and environmental temperature variation on the elastic vibration frequency and damping ratio is investigated numerically for different boundary conditions. The numerical results show that, under different boundary conditions, the compressive thermal stresses lead to the decrease of nominal vibration frequencies, and the increase of critical viscos coefficient and the damping ratio. With the increase of κ and the decrease of ξ , the damping ratio increases and decrease for strain-driven (εD) and stress-driven (σD) nonlocal models, respectively.