A general dynamic theoretical model of elastic micro-structures with consideration of couple stress effects and its application in mechanical analysis of size-dependent properties

Abstract

A general and systematic theoretical framework of elastic micro-structures is established with the aid of modified couple stress theory for investigating the size-dependent property in small scale, in which the size-dependence is considered by introducing a material length scale parameter. Mathematically, dynamic governing equations and corresponding boundary conditions are derived and simplified by using single power series expansion for a micro-plate and double power series expansion for a micro-beam. It is demonstrated that this method exhibits extraordinary superiority, i.e., different vibration modes can be extracted easily from artificial truncations. This theoretical model can be reduced to some classical cases, including the Bernoulli–Euler beam, Timoshenko beam, Kirchhoff plate and Mindlin plate, if some specific assumptions are made. After validation, a systematic numerical investigation is carried out, which focuses on the couple stress effect on shear resonance of a cantilever micro-plate. Finally, a methodology for proposing the critical size that distinguishes micro-scale from macro-scale is illustrated in detail.