## 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

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

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.

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

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

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

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

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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