An efficient semi-analytical formulation for the Lamb-like waves in layered waveguides based on global discretization

Abstract

Lamb-like waves are widely used in the health monitoring of layered structures. It is well known, however, that the semi-analytical finite element method, being capable to analyze the wave characteristics, requires a refined mesh, while the mode expansion technique requires a large number of modes in order to approximate the continuity conditions at the interfaces as well as the boundary conditions of the waveguide. In the present work, an efficient semi-analytical formulation is proposed based on the global discretization (GDSA), where all the continuity and boundary conditions can be exactly satisfied. First, the characteristic equation of an elementary layer is derived based on the wave structure described by the superposition of a polynomial and a series of sine basis functions, and the involved matrices are explicitly derived in form of the Kronecker product. Layered waveguides can then be modelled from the superposition of each layer. Based on this, a general transform, allowing to eliminate the imaginary parts of the eigenvalue problem, is proposed, and a reduced-order eigenvalue problem is developed which is capable to predict a majority of the cut-off frequencies. From the benchmarks on an aluminum plate and the plate covered by an ice layer, the convergence, accuracy and efficiency of the GDSA have been confirmed by comparing to existing methods. Lastly, the wave responses simulated from the spectral element analysis are taken for a case study. It is additionally found that the cut-off frequency of the A1 mode and the conversion between the symmetric and anti-symmetric modes are sensitive to the ice layer, having potential to the applications of ice detections.