Surface energy-enriched gradient elastic Kirchhoff plate model and a novel weak-form solution scheme

In this work, we propose a non-classical Kirchhoff plate model to investigate surface energy and gradient elasticity effects on the static bending and free vibration behavior of micro-plates. Gurtin-Murdoch surface elasticity theory and a single-parameter gradient elasticity theory are combined to capture three types of size effects. The equations of motion and related boundary conditions of the model are obtained by the energy variational principle. A C²-type differential quadrature finite element is constructed to solve the resulting sixth-order boundary value problem of micro-plates. Theoretical model validation and solution method verification are made through comparison with the available literature. Finally, the new plate model is applied to analyze the mechanical behavior of micro-plates and to carry out detailed parametric studies. Our results demonstrate that the combined effects of three physical reasons can result in not only the stiffening or softening behavior of static deflections and vibration frequencies but also the significant change in the vibration mode shapes of micro-plates. Further, the introduction of strain gradient effect can introduce a boundary layer at the simply supported edge.