Boundary Value Problems of a Kirchhoff Type Plate Model Based on the Simplified Strain Gradient Elasticity and the Application

Abstract

A new type of thin plate model and the related nonclassical boundary value problems were established within the framework of strain gradient and velocity gradient elasticity. The closed-form solutions of deflections and free vibrational frequencies of a simply supported plate resting on an elastic foundation were obtained. The results of the present model agree well with those predicted by the molecular dynamics. Numerical results show that, the elastic foundation and the strain gradient parameter have a stiffness-hardening effect, while the velocity gradient parameter has a stiffness-softening effect. The proposed boundary value problems are of great significance to the study of the mechanical behaviors of plates under complex boundary conditions and external loadings. Furthermore, it will be useful for developing effective numerical methods such as the finite element method, the finite difference method and the Garlerkin method.