Complex dispersion relations and evanescent waves in periodic beams via the extended differential quadrature method

Because of computation difficulties, investigations about the complex band structures and the evanescent wave modes in phononic crystals are very limited. In this paper, a novel k(omega) method, referred to as the Extended Differential Quadrature Element Method (EDQEM), is successfully developed to investigate the complex dispersion relations and the evanescent wave modes in periodic beams. At first, based on the Bloch-Floquet theorem and the two widely used beam theories, i.e., the Euler-Bernoulli beam theory and the Timoshenko beam theory, the EDQEM is developed to solve the dispersion equations of flexural waves in periodic beams. Comparisons with other related investigations are conducted to validate the correctness of the proposed method. Furthermore, considering three important factors, the shape of the unit cell, the pattern of the sampling point as well as the number of the sampling point, the convergence of the proposed method is investigated. Second, with the help of the EDQEM, complex dispersion relations of periodic beams are investigated and wave mode analysis is conducted, from which all possible waves, including propagative waves, purely evanescent waves and complex waves, in the complex dispersion curves are discussed. It is found that complex wave modes in periodic beams arise from two situations: (1) at the boundary of the first Brillouin zone and (2) within the first Brillouin zone. These complex wave modes are the transition modes between two propagative waves, or between the propagative wave and the complex wave. When the damping effect is included, all waves in periodic beams transfer into the complex waves. And, band gaps are not truly apparent anymore.