Free Vibration of Axially Moving Functionally Graded Nanoplates Based on the Nonlocal Strain Gradient Theory

Abstract

Using the nonlocal strain gradient theory, we explore vibration behaviours of initially bidirectional tensioned functionally graded nanoplates with axial speed. The governing equation of motion can be obtained based on the differential type of nonlocal strain gradient constitutive relation, which characters the dynamics of nanostructures containing kinematic relation. The simply supported boundary constraints on four sides are considered and subsequently the numerical results are determined. It shows that natural frequencies of axially moving nanoplates decrease when increasing the axial speed and the nonlocal parameter. Hence the nonlocal and kinematic factors cause the natural frequencies to decrease or, weaken the equivalent bending rigidity. On the other hand, natural frequencies increase with an increase in the axial tension and material characteristic scale. Hence the strain gradient and tensile stress factors cause the natural frequencies to increase or, strengthen the equivalent bending rigidity. In addition, the natural frequencies get higher with a larger aspect ratio of the functionally graded nanoplate. The larger one between the nonlocal parameter and the material characteristic scale plays a dominant role in the softening and stiffening mechanisms of the nonlocal strain gradient effect. In case of the same magnitude of the nonlocal parameter and the material characteristic scale, the softening and hardening phenomena disappear. The equivalent bending rigidity neither increases nor decreases in such a situation, and its value degenerates to the classical one.