On governing equations for a nanoplate derived from the 3D gradient theory of elasticity

The paper is concerned with the asymptotically consistent theory of nanoscale plates capturing the spatial nonlocal effects. The three-dimensional (3D) elasticity equations for a thin plate are used as the governing equations. In the general case, the plate is acted upon by dynamic body forces varying in the thickness direction, and by variable surface forces. The thickness of the plate is assumed to be greater than the characteristic micro/nanoscale measure and much smaller than the in-plane characteristic dimension (e.g., the wave or deformation length). The 3D constitutive equations of gradient elasticity are used to link the fields of nonlocal stresses and strains. Using the asymptotic approach, a sequence of relations for stresses and displacements in the form of polynomials in the transverse coordinate with coefficients depending on time and in-plane coordinates was obtained. The asymptotically consistent 2D differential equation governing vibration (or static deformation) of a plate accounting for both transverse shears and the spatial nonlocal contribution of the stress and strain fields was derived. It was revealed that capturing nonlocal effects in all directions leads to an increase in the correction factor compared with the well-known 2D theories based on kinematic hypotheses and the Eringen-type gradient constitutive equations. The effect of the internal length scales parameters on free low-frequency vibrations and displacements of a plate is discoursed.