Size-dependent topology optimization for eigenfrequency maximization of microplates using consistent couple stress theory

Abstract

This paper aims to maximize the fundamental eigenfrequency of the Mindlin microplate using the topology optimization approach and consistent couple stress theory (CCST). The modified bound formulation and the modified solid isotropic material with penalization (SIMP) approach are implemented to circumvent some undesired issues including repeated eigenfrequencies and localized mode. A four-node finite element with 80 degrees of freedom is used to meet the CCST-based C¹ continuity requirements, which reduces to 48 degrees of freedom when stretching deformation is neglected. The generalized optimality criteria method (G-OCM) is adopted to update the design variables. Subsequently, undesired frequency bands are removed by using a continuously differentiable frequency band constraint (FBC). The optimal topology for in-plane and out-of-plane vibration as well as their coupling mode are presented. Also, the optimal distribution of the nano-reinforcement phase to maximize the eigenfrequency of the nanocomposite microplate is determined. To achieve this, the reinforcement-phase element volume fraction and a special unified homogenization function, instead of SIMP, are considered for the design variable and material properties interpolation scheme, respectively. The small-scale effects are investigated by using some benchmark examples, which reveal that as in-plane dimensions approach the length scale parameter, the optimal topology will no longer resemble the classical form. Moreover, the obtained results show that the proposed nano-reinforcement design significantly increases the microplate fundamental eigenfrequency by the addition of only 1 % of reduced graphene oxide (rGO).