Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects

A new model for determining band gaps for wave propagation in two-dimensional (2-D) periodic composite structures is developed using a modified couple stress theory. The general equation of motion and boundary conditions in the elasto-dynamics of the modified couple stress theory are first derived by a variational formulation based on Hamilton's principle. The in-plane and anti-plane wave equations incorporating microstructure effects are then obtained explicitly from the general equation of motion. The plane wave expansion method and the Bloch theorem for periodic media are used to solve the in-plane and anti-plane wave equations, which are reduced to an eigenvalue problem in each case. The band gaps are determined from solving the characteristic equation and plotting the resulting eigen-frequencies. The new model recovers the classical elasticity-based model when microstructure effects are not considered. To quantitatively illustrate the newly developed model, a parametric study is conducted for 2-D periodic composite structures containing circular and square inclusions. The numerical results reveal that the microstructure effects on the band gaps are significant only when the unit cell size is small for both the composite structures. In addition, it is found that the volume fraction has a significant effect on the band gap size, and the inclusion shape has a large influence on the band gaps.