New closed-form solutions for flexural vibration of microplates based on a modified strain gradient theory

In this work, the flexural vibration of size-dependent Kirchhoff microplates has been investigated analytically following a modified strain gradient theory. A displacement-based variational formulation is established to derive the governing differential equations and all possible boundary conditions, including both essential and natural ones, for general curved boundaries. In particular, the natural boundary conditions are expressed in terms of force quantities in a concise form. The direct separation of variables method (DSVM) is developed to handle the eigenvalue problems of rectangular microplates with any combination of simply supported and clamped boundaries. Especially for the cases wherein two adjacent sides are clamped and the remaining sides are either simply supported or clamped, the closed-form solutions which used to be recognized as inexistent, are successfully obtained in the current work for the first time. These newly obtained results can serve as benchmarks for other relevant researches. Additionally, a parametric analysis is conducted to assess the influences of material length scale parameters and boundary conditions on size-dependent behavior of microplates.