## Abstract

The main cause of train derailment is related to transverse defects that arise in the railhead. These consist typically of opened or internal flaws that develop generally in a plane that is orthogonal to the rail direction. Most of the actual inspection techniques of rails relay on eddy currents, electromagnetic induction, and ultrasounds. Ultrasounds based testing is performed according to the excitation-echo procedure [1]. It is conducted conventionally by using a contact excitation probe that rolls on the railhead or by a contact-less system using a laser as excitation and air-coupled acoustic sensors for wave reception. The ratio of false predictions either positive or negative is yet too high due to the low accuracy of the actual devices. The inspection rate is also late; new numerical method has been developed in this context: The semi-analytical finite element method SAFE. This method has been applied in the case of anisotropic media [2], composite plates [3] and media in contact with fluids [4]. This method has been used successfully for several structures and especially in the case of beams of any cross-section such as rails that are the subject of this work and we were interested in wave propagation in waveguides of any arbitrary cross-section in the case of beams or rails.

## 1 Introduction

The propagation of elastic waves is widely used in the field of non-destructive testing [5], in particular for quality control and detection defect in mechanical components of machines and industrial installations. Inspection methods of rails that are based on ultrasonic wave propagation were widely used [6].

The SAFE approach to determine the dispersive curves is to discretize the domain cross-section by the finite element method, in a two-dimensional problem (2D). In the propagation direction of the wave, which is orthogonal to the cross-section, the displacements are modeled using harmonic analytic functions. Hence the name semi-analytical method of finite elements. The great merit of this approach is the reduction of computation time by comparison with a purely three-dimensional computation, and in particular for high frequencies or what amounts to the same at the small wavelengths [7].

## 2 Propagation of guided elastic waves in a rail

Let us consider an elastic waveguide with waves propagating along the direction

It is envisaged to search for the general solution of the waveguide in the form

The expression (2) expresses the separation between the movement in the plane of the cross-section and the off-plane motion which is considered purely harmonic. This requires that the material properties remain constant in the middle section of

By substituting Eq. (2) in Eq. (1), it comes

By explaining the calculation of derivation with respect to

To Eq. (4), we must add the boundary conditions on the free boundary of the right section which the stress on the boundaries are written:

The problem defined by Eq. (4) has the form of a quadratic problem with eigenvalues.

To formulate it discretely, the evaluation of spatial derivatives is necessary. The numerical approximation by the finite element method can be used to evaluate these derivatives.

The finite element method can be implemented directly from a variational principle such as the principle of virtual works. For a system subjected to the action of distribution of surface forces noted

For a free elastic system, the left part of the Eq. (6) vanishes:

In practice, it is perfectly possible to account for dissipation in the formulation of the variational problem via the introduction of elastic constitutive constants that are complex [9]. We then introduce the following star constants

The Eq. (7) becomes:

In the case of a slender system following the direction

We will note in the following the basic section which serves as a domain to the formulation SAFE by

The tensor of the deformations can be represented by a six-dimensional matrix. Using Eq. (10) and the definition of the small deformation tensor, we obtain the deformation field on the element

By substituting Eq. (13) for the elementary integral that appears in Eq. (7), we get:

By assembly, the discretized problem is written

Here the following global assembled system of dimension is equal to the total number of nodes

It is possible to make a base change in order to eliminate the pure imaginary complex that appears in Eq. (19). The following transformation matrix is thus introduced [7]

The new matrix

The new quadratic problem with the proper values to be solved in

The eigenvalue problem (22) can be solved by fixing the frequency

The quadratic problem (22) of the eigenvalue to be solved by writing in the following form

The matrices

From a numerical point of view, the eigenvalue problem defined by Eq. (18) or (21) is more convenient than the problem (23). The latter is slower than the first and is justified only in the case of depreciation in the structure [10].

The eigenvalue problem (23) is solved by specific numerical methods. The Matlab command *polyeig* solves it with the following command:

The eigenvalue problem (23) allows parametric definition for each value of

## 3 Numerical calculation of the forced regime solution

For a forced system in the context of the SAFE formulation, the stress vector is interpolated in the harmonic form

Considering the spectral component

By substituting Eq. (27) in Eq. (6) and using Eqs. (16) and (17), it comes by assembling

After assembly and multiplication on the left by the matrix

Using Eq. (23), we obtain

The solution of Eq. (30) is written

As the displacement represents the lower part of the vector

Considering a load punctual in

Nodal displacement in physical space is then given by

Then, by interpolation, we obtain the harmonic displacement field in the form

Note that calculating the displacement field by Eqs. (35) and (36) is a hard job.

It is necessary to determine the vector

## 4 Numerical calculation of the group velocity of the guided elastic waves in a rail

It is possible to implement the finite element method to calculate the global matrices

The group velocity that characterizes the shape of the wave is defined by

In the conservative case, it is possible to explain the velocity group for each particular solution

Cutoff frequencies are obtained by posing

In the case of a damping system for which the coefficients

## 5 Conclusion

This generalist method applies to any beam type waveguide and allows to parametrically analyze the various possibilities of excitation of the structure able to highlight targeted defects present in the structure. The semi-analytical finite element method offers the possibility of calculating the displacement field and the reflection and transmission coefficients when a certain defect is considered on the cross-section of the rail.

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