Traveling and standing flexural waves in the micro-beam based on the fraction order nonlocal strain gradient theory

The traveling and standing flexural waves in the micro-beam are studied based on the fraction order nonlocal strain gradient elasticity in the present paper. First, the Hamilton's variational principle is used to derive the governing equations and the boundary conditions with consideration of both the nonlocal effects and the strain gradient effects. The fraction order derivative instead of the integer order derivative is introduced to make the constitutive model more flexible while the integer order constitutive model can be recovered as a special case. Then, the Euler-Bernoulli beam and the Timoshenko beam are both considered and the corresponding formulations for them are derived. Two problems are investigated: 1) the dispersion of traveling flexural waves and the attenuation of the standing waves in the infinite beam. 2) The natural frequency of finite beam. The numerical examples are provided and the effects of the nonlocal and the strain gradient effects are discussed. The influences of the fraction order parameters on the wave motion and vibration behavior are mainly studied. It is found that the strain gradient effects and the nonlocal effect have opposite influences on the wave motion and vibration behavior. The fraction order also has evident influence on the wave motion and vibration behavior and thus can refine the prediction of the model.