Analysis of wave propagation in beams with transverse and lateral cracks using a weakly formulated spectral method

In this paper, a novel numerical technique based on the global-local hybrid spectral element (HSE) method is proposed to study wave propagation in beams containing damages in the form of transverse and lateral cracks. The ordinary spectral element method is employed to model the exterior or far field regions, while a new type of element (HSE) is constructed to model the interior region containing damages. To develop this efficient new element for the damaged area, first, the flexural and the shear wave numbers are explicitly determined using the first-order shear deformation theory. These wave modes, in one of the two mutually orthogonal directions for two-dimensional transient elastodynamics, are then used to enrich the Lagrangian interpolation functions in context of displacement-based finite element. The equilibrium equation is then derived through the weak form in the frequency domain. Frequency-dependent stiffness and mass matrices can be accurately formed in this manner with a coarse discretization. The proposed method takes the advantage of using (i) a strong form for one-dimensional wave propagation and also (ii) a weak form by which a complex geometry can be discretized. Numerical verification is carried out to illustrate the effectiveness of the method. Finally, this method is employed to investigate the behaviors of wave propagation in beams containing various types of damages, such as multiple transverse cracks and lateral cracks.