Investigation of size effect on band structure of 2D nano-scale phononic crystal based on nonlocal strain gradient theory

Abstract

Phononic crystal (PnC) has attracted strong attention due to its tremendous capabilities to manipulate acoustic and elastodynamic waves. For high frequency (gigahertz) PnC, the size effect becomes significant when the dimension of the structure approaches nano-scale. In the present paper, the size-dependent band structure of a two dimensional (2D) PnC is studied by utilizing differential governing equation of nonlocal strain gradient theory (NSGT). The general equation of motion and boundary conditions are first derived by a variational formulation based on Hamilton's principle. Based on general form partial differential equation module in COMSOL® Multiphysics, a method is further proposed to solve the non-classic wave equations, which are derived from strain-, and stress-driven nonlocal models, and NSGT. The square lattice PnCs with circle, cross, and octangle air holes are investigated. Numerical results show size effect becomes significant when the lattice constant approaches nano-scale. The smaller size results in the stronger size effect. The influence of two nonlocal parameters on the first band gap is investigated simultaneously. The band gap always tends to narrower because of the size effect. Finally, it is found that the strength of size effect is mainly related to the thickness of minimum connector, but not to the filling ratio and surface length. This study paves the way for the studying and designing PnC in nano-scale or at ultra-high frequency.