Dynamic Response of Fractional-Order Viscoelastic High-Order Shear Beam Based on Nonlocal Strain Gradient Elasticity

The dynamic behavior of a viscoelastic high-order shear microbeam is studied based on a new constitutive model which incorporates size effects and viscoelasticity simultaneously. The size effects are modeled by the nonlocal gradient elasticity, while viscoelastic effects are modeled by fractional-order derivatives. The constitutive relation and the equations of motion are both differential equations with fractional-order derivatives. Based on the Laplace transform and inverse transform, the analytical solution of the dynamic response under a step load is obtained in terms of the Mittag–Leffler function. In order to verify the reliability of the analytical solution, a comparison with the numerical solution is also provided. Based on the numerical results, the effects of the nonlocal parameter, strain gradient parameter, fractional-order parameter, and viscosity coefficient on the dynamic response of the viscoelastic microbeam are discussed. It is found that the influences of the fractional order and the coefficient of viscosity on the dynamic response of the microbeam are very different, although both are related to the viscoelastic behavior.