Time-domain computation of the response of composite layered anisotropic plates to a localized source
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 in the plate plane. By doing so, we obtain a partial differential equation in a plane perpendicular to direction , and consequently to the plate too, for which the angular frequency and the wavenumber in direction 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 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 , 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 -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 -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 of perfect flat layers of normal , stacked together. Each layer is an anisotropic solid, with a given thickness , as illustrated in Fig. 1. The total thickness is expressed by . Above and below this plate system are semi-infinite vacuum half-spaces. The plate is assumed to be infinite in the -plane, where the position is denoted by the vector .
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 , and (e.g., [5], [16], [6]). The three space and time frequencies, denoted by and , 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 and for each partial mode , the wave number in the -direction is then related to the three variables and 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
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Lamb waves in the wavenumber–time domain: Separation of established and non-established regimes
2021, Wave MotionCitation 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, UltrasonicsCitation 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].