Stability of Vibrating Functionally Graded Nanoplates with Axial Motion Based on the Nonlocal Strain Gradient Theory

Abstract

Considering both the nonlocal scale and material length scale effects, we investigate vibration and stability behaviors of functionally graded nanoplates with axial motion in order to model the two-dimensional nanobelt in nanoengineering. The nonlocal strain gradient theory is applied and the differential nonlocal strain gradient constitutive relation is adopted. Using the physical neutral plane of a functionally graded thin plate, we derive the governing equation of motion for the functionally graded nanoplate with axial motion via Hamilton’s principle, where the kinematic characteristics are introduced into the dynamic behaviors. The governing equation is numerically solved using the Galerkin method. Effects of the nonlocal scale and material length scale parameters, axial velocity, gradient index, biaxial pre-tensions and aspect ratio are discussed. The results demonstrate that complex frequencies of the functionally graded nanoplate with axial motion decrease with an increase of axial velocity in the subcritical region, while the moving nanoplate experiences a divergent instability or flutter instability in the supercritical region. Natural frequency and critical speed decrease with the increase of the nonlocal scale parameter while increase with the increase of the material length scale parameter, reflecting the nonlocal softening and strain gradient hardening mechanisms, respectively. Besides, natural frequency and critical speed increase with the increase of the biaxial pre-tensions and aspect ratio, but decrease with the increase of the gradient index. In particular, the influences of the gradient index and size or weight of functionally graded nanoplates on the critical speed are explored.