Wave propagation in periodic nano structures through second strain gradient

The study of wave propagation in the periodic nano-waveguides is of importance to advance the development of energy and wave controlling systems. In this paper, firstly, the element discretization of nano structures is introduced based on the second strain gradient theory. The displacement field is determined through the utilization of six quintic Hermite polynomial interpolating functions. By applying Hamilton's principle, the weak form, which incorporates the element matrices, is determined and the global dynamic stiffness matrix of a unit cell is assembled. Then, the wave finite element method, based on the inverse form, direct form and contour integral solution, is used to analyze the two-dimensional free wave propagation properties of complex nano structures by solving the eigenvalue problems. The interpretation of the frequency spectrum is confined to the irreducible first Brillouin zone, encompassing both the dispersion relation and band structure. Furthermore, the slowness surfaces and energy flow vector fields are discussed via the second strain gradient theory. The novel work, which utilizes the second strain gradient theory equipped with the wave finite element method to investigate two-dimensional complex nano structures, has revealed significant potential in elucidating the two-dimensional wave propagation characteristics of the complex nano-waveguides.