A general dynamic model based on Mindlin's high-frequency theory and the microstructure effect

摘要

Based on Hamilton's principle, a general three-dimensional mechanical model with consideration of the consistent couple stress theory is established in this paper, and the dynamic governing equations and boundary conditions are derived strictly. After that, inspired by the high-frequency theory established by Mindlin, a novel two-dimensional theory of micro-plates is proposed. Governing equations and boundary conditions are simplified and further validated via some reductions and comparisons with the traditional Mindlin plate and the Timoshenko beam. As an application, the propagation property of straight-crested waves in a micro-plate is investigated, and the effect of couple stress is revealed. Then, the static bending and free vibration of a simply supported rectangular micro-plate is investigated with the aid of the double Fourier series. It is demonstrated that the couple stress has significant influence on the mechanical behaviors, including the static deflection and natural frequency, as the plate thickness shrinks. Physically, the couple stress effect is equivalent to an increase in structural stiffness. In order to illustrate the effect of the couple stress, a general criterion for quantifying the critical size that distinguishes microscale from macroscale is proposed numerically. To some extent, it can also be viewed as a criterion for distinguishing the consistent couple stress theory of micro-structures from the classical continuum mechanics theory of macro-structures.