Elsevier

Wave Motion

Volume 51, Issue 8, December 2014, Pages 1364-1381
Wave Motion

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

https://doi.org/10.1016/j.wavemoti.2014.08.003Get rights and content

Highlights

  • Response of a layered anisotropic plate to an ultrasonic source calculated in the time-domain.

  • Independent one-dimensional vibration problems to solve.

  • Eigenfrequency and shape of a Lamb mode calculated with respect to its wave-vector.

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 n-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.

Introduction

The study of the diffraction of an ultrasonic source in an anisotropic multi-layered plate is of great interest in the field of Nondestructive Testing. From the point of view of propagation analysis, knowing the amplitude distribution of the elastodynamic field in different directions and studying ultrasonic wave propagation to the area of inspection are both important, for example, to choose the correct source for a specific problem under consideration. This is mainly true for inspections of composite materials, which are generally made of oriented fiber stacks. Their acoustic responses are complex and most of the time strongly anisotropic.

There are different harmonic methods to calculate the diffracted field, at any point and any time, in such plates (e.g.,[1], [2]). The method most commonly used, without doubt, consists of applying a double Fourier transform to the initial partial differential equation in four variables, with respect to time and one direction d in the plate plane. By doing so, we obtain a partial differential equation in a plane perpendicular to direction d, and consequently to the plate too, for which the angular frequency and the wavenumber in direction d are parameters (see for example  [3], [4]). The solution is then decomposed into guided modes, i.e. generalized Lamb waves, and the wavenumber in the direction along the plate perpendicular to direction d is calculated from all the other spectrum variables. The guided wave can then be calculated by using a global matrix  [5] for example, or an impedance matrix  [6]. These methods are semi-analytic. Pure numerical methods exist like the SAFE method ([7], and e.g.,  [8]), which can be used to model very complex plane structures. When the modes are known, the residue theorem, or other techniques such as those based on the reciprocity theorem, can be applied to calculate mode amplitude (e.g.,  [3]). These field calculations are involved when developing hybrid methods, which consist of coupling mode theory with finite element methods  [9], [8]. In essence, these approaches are well adapted to calculate the field for long distances of propagation along the plate, since the solution is given in terms of Lamb waves. In contrast, they are not so efficient for analyzing the field for sub-source locations, because a very large number of evanescent modes are involved in the field description.

There is an alternative technique, however, which is to do the calculation in the time-domain. Strangely, few studies can be found in the literature on this subject  [10], [11]. In this case, a 2D Fourier transform is performed with respect to the coordinates in the plane of the layers (and of the interfaces) on the partial differential equation. In this 2D spectrum domain, i.e. at any fixed wave-vector k, the solution is obtained by summation of plate resonances. A good introduction can be found in  [10], showing that this method has been developed and exploited in geophysics. In this paper, we propose to develop this approach to study wave propagation in composite materials for Nondestructive Testing applications.

After a brief description of the basic equations (Section  2), using our notations, the solution is obtained in the time-domain (Section  3), analogously to what has been done by the so-called “Thin-Layer Method”  [10], except that there is no discretization in the z-direction perpendicular to the plate. Indeed, in the Thin-Layer Method, “to solve the wave equation, [one] begins by dividing the physical domain in layers that are thin in the finite element sense […]”  [10]. Mode shapes are defined here analytically for the most general case as continuous functions of the z-coordinate, similarly to what is suggested for the isotropic case in  [11]. Nevertheless, at the last stage of the calculation, either semi-analytical or pure numerical techniques must be used to calculate the resonances of the plate. The finite difference method is used here for its efficiency but it is not necessary (see for example  [5], [6] in the frequency-domain). In Section  4, the numerical aspects are presented and finally, in Section  5, different results are shown for two typical plates. It will be emphasized that the method is accurate for observation points located both under the source and reasonably far from it along the plate.

Section snippets

Equations in the physical domain

Let us consider a multilayered medium consisting of a plate system with a number m of perfect flat layers of normal z, stacked together. Each layer is an anisotropic solid, with a given thickness hβ=zβzβ1, as illustrated in Fig. 1. The total thickness is expressed by h. Above and below this plate system are semi-infinite vacuum half-spaces. The plate is assumed to be infinite in the xy-plane, where the position is denoted by the vector x=(x,y).

This stratified medium is subjected to an

An infinity of one-dimensional vibrating systems

The usual method used to solve Eqs. (3), (4), (5) is to apply a 3D Fourier transformation to them, with respect to x, y and t (e.g.,  [5], [16], [6]). The three space and time frequencies, denoted by kx,ky and ω, are independent variables. The first two are the components of the wave vector on the surface and the last one is the angular frequency. In each layer β and for each partial mode i, the wave number κβ,i in the z-direction is then related to the three variables kx,ky and ω by the

Computation

Once a rectangular domain of observation and discretization steps have been chosen, the computation is performed in two successive stages: firstly, for a given layered plate, the different modes are calculated and stored. The mode computation has to be done once and for all. Secondly, the time-domain response of the plate to a given solicitation is calculated.

Numerical results

This section reports some numerical results based on our numerical schemes. Two typical plates are analyzed. The first plate is made of a single layer of a unidirectional fiber composite. The second is a structure made of a unidirectional fiber composite plate and a steel plate stuck together.

Conclusion and future prospects

The method that has been developed in this work, based on time-domain analysis, permits to calculate ultrasonic source diffractions in multilayered composites. This method is a useful tool which complements the most commonly used technique based on generalized Lamb wave decomposition, for which precise calculations can be made for long propagation distances, in a relatively short time. However, the most interesting feature of the modal decomposition in the time-domain is the fact that it is

References (29)

  • L. Taupin, A. Lhémery, V. Baronian, A.-S. Bonnet-BenDhia, B. Petitjean, Hybrid SAFE/FE model for the scattering of...
  • L. Taupin et al.

    Scattering of obliquely incident guided waves by a stiffener bonded to a plate

    J. Phys.: Conf. Ser.

    (2012)
  • E. Kausel

    Thin-layer method: formulation in the time domain

    Internat. J. Numer. Methods Engrg.

    (1994)
  • J. Park et al.

    Response of layered half-space obtained directly in the time domain, part I: SH sources

    Bull. Seismol. Soc. Am.

    (2006)
  • Cited by (4)

    • Mode computation of immersed multilayer plates by solving an eigenvalue problem

      2022, Wave Motion
      Citation Excerpt :

      Specifically for nonradiating waveguides, various methods have been developed over the last decades to make this discretization. As examples of such methods, let us cite the following: the thin-layer method (TLM) [3,4], the semi-analytical finite element (SAFE) approach [5], the scaled boundary finite element method (SBFEM) [6], the high-order finite difference scheme [7], spectral collocation [8,9], the complex-length finite element method (CFEM) [10], or decomposition on basis functions [11]. For immersed or embedded one-dimensional waveguides, such as rods and pipes, two main approaches have been used to obtain an eigenvalue problem.

    • Lamb waves in the wavenumber–time domain: Separation of established and non-established regimes

      2021, Wave Motion
      Citation Excerpt :

      We briefly recall in this section the expression of the Green’s tensor in terms of the Lamb modes obtained in the wavenumber domain. Thorough derivations can be found in [6,7]. Let us now go to an example.

    • Transient 3D elastodynamic field in an embedded multilayered anisotropic plate

      2016, Ultrasonics
      Citation Excerpt :

      This is because in the far field the modes are well-built, and only a limited number of them, i.e., the propagative modes, contribute, which makes modal theory very efficient. For plates in vacuum, the solution can also be expanded in terms of modes in the time domain, after performing a Fourier transform with respect to the horizontal coordinates (in the plane of the plate) [12,7], assuming that the source is bounded and bandlimited in horizontal wavenumber, according to the Nyquist–Shannon sampling theorem. This second method is more suitable than the first one for 3D calculations in anisotropic media because it does not make use of an angular integration variable, nor is the direction of propagation of the modes required [26].

    View full text