Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 475364, 12 pages

http://dx.doi.org/10.1155/2015/475364

## Floquet-Bloch Theory and Its Application to the Dispersion Curves of Nonperiodic Layered Systems

^{1}Hydrogeophysics and NDT Modelling Unit, University of Oviedo, C/Gonzalo Gutiérrez Quirós s/n, 33600 Mieres, Spain^{2}Dynamics Division, Applied Mechanics Department, Chalmers University of Technology, Hörsalsvägen 7, 41296 Gothenburg, Sweden

Received 20 October 2014; Revised 28 November 2014; Accepted 29 November 2014

Academic Editor: Xiao-Qiao He

Copyright © 2015 Pablo Gómez García and José-Paulino Fernández-Álvarez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Dispersion curves play a relevant role in nondestructive testing. They provide estimations of the elastic and geometrical parameters from experiments and offer a better perspective to explain the wave field behavior inside bodies. They are obtained by different methods. The Floquet-Bloch theory is presented as an alternative to them. The method is explained in an intuitive manner; it is compared to other frequently employed techniques, like searching root based algorithms or the multichannel analysis of surface waves methodology, and finally applied to fit the results of a real experiment. The Floquet-Bloch strategy computes the solution on a unit cell, whose influence is studied here. It is implemented in commercially finite element software and increasing the number of layers of the system does not bring additional numerical difficulties. The lateral unboundedness of the layers is implicitly taken care of, without having to resort to artificial extensions of the modelling domain designed to produce damping as happens with perfectly matched layers or absorbing regions. The study is performed for the single layer case and the results indicate that for unit cell aspect ratios under 0.2 accurate dispersion curves are obtained. The method is finally used to estimate the elastic parameters of a real steel slab.

#### 1. Introduction

Floquet-Bloch (hereafter F-B) theory provides a strategy to analyze the behavior of systems with a periodic structure. Floquet’s seminal paper dealt with the solution of 1D partial differential equations with periodic coefficients [1]. In solid state physics, Bloch generalized Floquet’s results to 3D systems and obtained the description of the wave function associated with an electron traveling across a periodic crystal lattice [2]. This wave function is a solution of the Schrödinger equation with a periodic potential and Bloch showed that it was the product of a simple plane wave multiplied by a periodic function with the same periodicity of the lattice. The mathematical description of these ideas, in the context of quantum mechanics, can be found in [3, 4].

In the literature dealing with wave propagation problems in mechanical systems the theory is referred to as Floquet-Bloch theory or, simply, Floquet theory. In layered systems, due to the heterogeneity of the relevant elastic properties, to particular geometric features, or to both, only certain wave modes can physically propagate inside the structure [5]. Each of these modes can be identified by a determined—generally nonlinear—function relating the time frequency and the spatial frequency (or wave number). These relationships are called* dispersion curves* default and, as they summarize all the oscillatory behavior of the system, their calculation is of paramount importance in NDE applications [6].

Vibrations occur also in objects with periodic structure [7]. These problems usually admit a separation between the time and the spatial dependent parts of the solution. For instance, the Helmholtz equation is a known example of equation describing the spatial behavior [8]. There, the physical periodic structure of the studied object translates into spatial periodicity of its coefficients. Therefore, the F-B theory has been applied to obtain the dispersive properties of different mechanical periodic systems [8–12].

Many relevant structures can be assumed to be layered systems of infinite extent, for example, [13, 14] in civil engineering constructions, [15, 16] in optics, or [17] in electromagnetics. Therefore, theoretical methods and experimental techniques to obtain their dispersion curves have been devised. From the theoretical side, different matrix techniques have been developed to address the calculation. They involve numerical computational methods whose complexity increases with the number of layers in the system [18, 19] or more recently [6].

In laboratory experiments or field work, the dispersion curves can be obtained using, for example, the multichannel analysis of surface waves (MASW) method. The MASW procedure involves collecting equally spaced measures of vibration along a profile on the system surface using, for example, accelerometers. The resulting 2D space-time discrete image is Fourier-transformed to the frequency-wave number domain and then processed to build the dispersion curves [20, 21]. The method has some drawbacks inherent to the Fourier transform limitations which will be discussed later. The MASW has been applied successfully in the characterization of pavement systems [13], as a seismic data acquisition technique [20] or for geotechnical characterization [22]. The MASW strategy is here also used to perform a computer numerical simulation of the system, closely mimicking the field setup. The issue of infinite lateral extent is usually tackled by using perfectly matched layers (PML) [23–26] as has been done here or absorbing regions. Both techniques present drawbacks [26].

In this paper, an alternative way to calculate the dispersion curves of layered systems with infinite lateral extent using the F-B theory is presented. The method has never been applied to the dispersion curves calculations of nonperiodic layered systems. Here it is used to obtain the dispersion curves of a single layer case and to estimate the elastic parameters of a real steel slab, for showing the method. However, the novelty in this work is that it can be applied to an arbitrary number of layers, even if the layers are anisotropic or orthotropic, with the same complexity level. The power of the method is that the equations are solved by the finite element software, because the F-B theory only affects the propagation term, which is the same, whatever the nature of the layers. It is not necessary to develop the equations for each specific problem and to generate complex codes to get the dispersion relations.

The F-B theory reduces the problem to calculations performed in the so-called unit cell, subject to certain specific boundary conditions derived from the F-B theory and elastodynamics. The influence of the size of the unit cell is ascertained. The results are first compared with the dispersion curves derived from the Rayleigh equations [5], solved by a searching root numerical method. Comparison is also made with the curves resulting from a FEM computer simulation followed by a 2D Fourier transformation. Finally, a real experiment was performed on a steel slab employing the MASW method and the empirical dispersion curves were compared with the analytical and numerical ones.

The results show that the F-B method compares favorably with other methods and fits accurately the empirical data, providing a good alternative to obtain dispersion curves in layered systems. The F-B technique can be run on a finite element package like COMSOL Multiphysics, can be applied to an arbitrary number of layers in the system, with the same complexity level, and eliminates issues of infinite lateral extent.

#### 2. Floquet-Bloch Theory: Explanation

The Floquet-Bloch theory provides a strategy to obtain a set of solutions of a linear ordinary equations system of the form where is the solution vector and the matrix is periodic such that for a certain period . At first sight it might seem that the solution of such a problem would have to be also -periodic. But Floquet showed that this need not be so. There exists, however, a simple relationship between the solution’s behaviors inside one period and outside it. If is a fundamental matrix of solutions, then another matrix can be found such that can be constructed by setting in (2) such that . A simplest case is obtained using so that . As there is not a unique choice for the fundamental matrix and how it is exactly chosen depends on the problem, is also not unique. But its eigenvalues are intrinsic of the problem and, under the right transformation, can be used as a propagator or evolution factor relating the value of the solution at a point inside the period with its value at a point outside of it. Only the solution inside a period is, therefore, needed verifying that Following the classical nomenclature is known as Floquet multiplier, being the complex Floquet exponent. Moreover, Floquet found that the solution at any point can also be factored in two terms: Here is a periodic function, playing the role of the eigenvectors if was a constant matrix and carrying the periodicity of the coefficients of the problem. The complex exponential distorts the strict periodicity of incorporating damping or ever growing effects in the amplitudes of the solution depending on the value of . This is why solutions are, in general, not periodic and also why the Floquet perspective is usually employed to study their stability. A solution will be stable if the Floquet multipliers verify [27].

#### 3. Guided Waves in Layers: Analytical Dispersion Curves

The guided wave propagation problem in a homogeneous, isotropic, and infinite single layer has been widely treated in the literature [5, 28]. In this paper, we follow the theory developed in [5]. So consider an infinite (in direction), homogeneous, isotropic, and elastic layer with thickness as shown in Figure 1.