Time-domain computation of the response of composite layered anisotropic plates to a localized source

Abstract

This paper describes how a modal approach in the time-domain can be suitable for calculating the elastodynamic field in a layered plate. This elastodynamic field is generated by impulsive sources located in a small region of a composite plate consisting of anisotropic layers stuck together. The aim is to calculate the transient response of the elastic plate around the location of the sources, generally emitting nn-cycle pulses. First, we apply a 2D Fourier transform to the wave equation with respect to the coordinates in the plate plane, and then, in the 2D spectrum domain, for any given wave-vector in the plate plane, solving a vibration problem with respect to time and position in the direction perpendicular to the plate. The solution is expressed as the sum of mode responses, each mode having a resonance frequency and a shape which depend on the wave-vector in the plate plane. These calculations are different from those obtained by the usual method in the harmonic domain, where the modes are searched for a fixed frequency, such as Lamb waves, i.e. guided waves that propagate along the plate. In our case, the solution is given as a summation of plate resonances, i.e. a decomposition on the real eigenfrequencies, associated to Lamb waves with the same fixed wave-vector. This difference is of importance since only Lamb modes with real frequencies and real-valued wavenumbers in the plate plane are involved here, contrary to the usual harmonic methods, where these modes can be evanescent. This is of great interest as it can simplify the calculation of the generated field near the source. Finally, we obtain a solution in the physical domain by performing an inverse 2D Fourier transform. After a detailed description of the method, results are shown for two typical plates. It is emphasized that the method is accurate for observation points located both above or below the source and reasonably far from it along the plate.