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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.

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 [

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 [

Vibrations occur also in objects with periodic structure [

Many relevant structures can be assumed to be layered systems of infinite extent, for example, [

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

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 [

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.

The Floquet-Bloch theory provides a strategy to obtain a set of solutions of a linear ordinary equations system of the form

The guided wave propagation problem in a homogeneous, isotropic, and infinite single layer has been widely treated in the literature [

An infinite homogenous isotropic and elastic layer with thickness

The conservation of momentum equation plus free traction boundary conditions leads to a system of equations that produces a solution when its determinant vanishes. In the absence of body forces the equation reads

The R-L equation is transcendental, so no closed analytical solution is available. It can, though, be cast into a form amenable to the use of iterative, root finding, local algorithms, but these present various difficulties arising from the nature of the equations. First, due to the tangent functions, the left hand side is discontinuous at certain points where local algorithms for smooth functions will find difficulties [

A visual inspection to the pattern followed by the dispersion curves on a phase velocity versus frequency

For local root searching methods, like the Newton-Raphson method [

In this paper, the root loci of the R-L equation have been obtained using the bisection method, which always converges. It is very slow and fails wherever multiple roots exist in the proposed interval. A pair of time frequency, phase velocity values are input in the R-L equation and its sign evaluated. Keeping the frequency fixed, the velocity has been varied with a step of

The elastic parameters input to the R-L equation are thickness

Some of the problems discussed above can be clearly seen in the

The multichannel analysis of surface waves (MASW) methodology is a procedure to numerically or empirically calculate dispersion curves. The strategy is to measure (or obtain numerically), on the surface of the system, the wave field in a number of equally spaced points and take readings at a certain temporal sampling rate [

Given the inverse relationship between the length of the profile on the surface and the wave number sampling interval and, correspondingly, between the range of sensed wave numbers and the distance between adjacent sensors [

When the scheme is applied to perform a numerical calculation on guided waves, the simulation domain has to be finite. This brings the problem of unwanted reflections and mode conversions at those boundaries, coming back into the relevant domain and corrupting the signal. The naive option of extending the domain implies increasing the number of nodes and computation times and assumes that boundary reflection events are separated in time from the studied events, which might not be possible. The use of the so-called absorbing regions, where the wave field enters and is computationally absorbed, has been treated in the literature [

In this case, COMSOL Multiphysics has been used for the simulation. The characteristic parameters employed for the PML are PML scaling factor

Typical scheme for MASW method implementation in a FEM software. The computational domain has been set to

In this section the results of the numerical simulations using the FEM software COMSOL Multiphysics, following a MASW procedure and employing PML to take lateral unboundedness into account, are presented. The dispersion curves will be discussed and serve as a reference to be compared with those obtained by the searching root algorithm (Section

A frequency domain study has been performed with a step of

Dispersion curves for a single layer in frequency-wave number representation (a) and in phase velocity-frequency representation (b). Parts that are lacking in the phase velocity plot (b) correspond to wave numbers greater than the measured ones.

Some observations are in order. First, certain portions of some modes are not excited [

The empty triangular space in the bottom right part of Figure

Additional numerical artifacts arise due to the finite length of the profile. As this is mathematically equivalent to multiplication by a boxcar window it generates, in the wave number domain, a convolution with a sinc function. This effect has been zoomed in Figure

Dispersion curves in

All the described difficulties with the MASW domain are absent in the Floquet-Bloch technique.

The equation of movement for the single layer case was presented in Section

Equation (

The layer is considered infinite; however, the solutions can be computed over a finite computational domain (unit cell), subject to certain boundary conditions. Consider an infinite layer as is shown in Figure

Infinite layer and the unit cell elected (a). Computational domain (unit cell) properties and dimensions (b). F-B boundary conditions applied to the laterals of the computational domain (unit cell) for a certain shape

According to [

Consider, now, the spatial part of (

Therefore, the displacement field at the left side of the unit cell (Figure

Note that the function

Moreover, a lateral infinite layered system is a trivially periodic medium in the propagation

For certain periodic function

From (

Now, the relation of the solutions at the left and right sides of the unit cell can be used to define the proposed F-B boundary conditions as

Due to the relationship between

The theory presented in this section applied to the lateral sides of the unit cell can be used for any kind of layered systems, whatever the nature of the layers (isotropic or anisotropic) and the number of them are. The reason is that the proposed F-B boundary conditions affect only the propagative part of the solution (

Therefore, the complicated part of the problem, which is to obtain the function

However, the perspective for getting analytical solutions is always the same, to obtain a function (complicated in general) in the thickness direction (the mode) which propagates laterally. In this approach, the propagation part is always extracted in the equations when infinite layered systems are considered.

Because of the above reason, the theory applied to define the F-B boundary conditions can be used for any complicated systems while they are laterally infinite. The boundary conditions only affect the propagation part and allow converting the infinity analytical problem, in a finite problem, which can be solved with commercial Finite Element software. Therefore, it is not necessary to develop equations (generally complicated) for each certain problem and perform complex numerical codes based on searching roots algorithms [

Using COMSOL Multiphysics software, the eigenvalue problem equation (

However, to transform the problem of the infinite layered system to a finite problem through the F-B boundary conditions, introduce artificial aspects due to the periodicity of the

An example is presented in Figure

Dispersion curves in

Outline of the experiment assembly (a), a photograph (b), and the impacts generated in the experiment in the time domain (c) and spectra (d). The average time has been

Since the calculation is developed in terms of the F-B wave vector, the solution becomes

Different aspect ratios of the unit cell

In the case

A good criterion is to choose the lateral dimension at least five times the thickness in the computational domain.

A real NDT experiment has been conducted on the surface of a steel slab shown in Figure

The shape of the hammer impacts in the time domain is very consistent (Figure

The resulting empirical dispersion curves can be seen in Figure

Superposition of the dispersion curves obtained with the numerical searching root method (

The estimated parameters using the proposed F-B boundary conditions are presented in Table

Estimated elastic parameters of a real steel slab using the experimental signals obtained with MASW method fitted with the proposed F-B method.

Density (kg/m^{3}) |
P wave velocity (m/s) | S wave velocity (m/s) |
---|---|---|

7850 | 5800 ± 30 | 3200 ± 23 |

The estimated errors are obtained with the propagation law (

A fit has been achieved where only the A0 mode is clearly seen in the measurements. There are two main reasons for that. On the one hand, the tip of the hammer only significantly inputs frequencies until 15 kHz (Figure

This section presents the comparison of the different methods to obtain dispersion curves: the searching root method, the numerical MASW method together with the proposed F-B method, and their ability to match the empirical dispersion curves. The results have been calculated for a layer with the same thickness as the slab used for the experiment. The numerical MASW simulation performed in COMSOL Multiphysics has used as impacts the experimental ones. Figure

From Figure

The main conclusion of this study is that the F-B theory can be used to compute the theoretical dispersion curves of layered systems with infinite lateral extension over a finite unit cell. The method is applied directly using a finite element commercial software and is free of the drawbacks associated with other numerical procedures used. It is also a tidy method to calculate curves with more than one layer, avoiding ambiguities at crossing points or with particular slopes. Based on the obtained results, aspect ratios with lower values than

The authors Pablo Gómez García and José-Paulino Fernandez-Álvarez declare that there is no conflict of interests regarding the publication of this paper.

Thanks are due to the Wave Propagation Division of Chalmers University and specially to Professor Anders Boström because of his suggested corrections in the development of this work.