Asymptotic long-wave model for an elastic isotropic nanoplate with surface effects derived from the 3D theory of elasticity and its comparison with hypotheses-based models

The paper deals with the derivation of asymptotically correct equations governing the long-wave bending response of a rectangular ultrathin elastic isotropic plate taking into account surface effects within the framework of the Gurtin-Murdoch theory of surface elasticity. The upper and lower faces are assumed to be pre-stressed by residual surface stresses which can be either tensile or compressive. The original 3D equations of elasticity are split into equations corresponding to the in-plane boundary layer and equations predicting out-of-plane bending deformation. By performing asymptotic integration through the thickness of the 3D equations associated with bending deformations and satisfying the balance equations on both faces, we derive asymptotically correct relations for displacements and stresses, as well as a new Timoshenko-type equation capturing surface stresses and inertia. A comparative analysis of the derived governing equation with similar available equations based on kinematic hypotheses revealed significant differences in the effective bending stiffness and factors of the inertia term. As examples, we studied free low-frequency vibrations and self-buckling of a square nanoplates made of different materials and compared effects of residual stresses on the natural frequencies and the critical value of the plate side using the novel model and the models relying on hypotheses for the normal component of the stress tensor.