Size-dependent microstructure design for maximal fundamental frequencies of structures

Topology optimization on a unit cell is a common technique to improve the fundamental frequencies of periodic cellular solid structures. During this procedure, the effective properties of cellular solids are primarily computed by the homogenization method. This homogenization method is based on the classic continuum theory under the assumption that the unit cell is infinitely small. Hence, this classic strategy is inadequate to interpret the size dependence of the optimal results. The aim of this study was to describe and examine size dependence in relation to the topology design of the unit cell to achieve maximization of the structural fundamental frequencies. For this purpose, we determined the effective properties of the cellular solids and constructed the optimization formulation based on the couple-stress theory rather than the classic theory. A modified bound formulation of the objective and constraint functions was used to avoid the non-differentiability of repeated frequencies. Although the existing theory does not reflect size dependence, our optimization formulation was able to identify the size dependence of both the microstructural topologies and the fundamental frequencies. The size-dependent results are achieved by varying of the mechanisms to achieve the maximal fundamental frequencies in response to cell size variation. The present formulation is suitable for the unit cell design of cellular solid structures that possess local dimensions comparable to the cell size, and this novel formulation has expanded the application scope of the classic microstructural design problem for periodic materials.