Slim Soua, Septimonette Chan and Tat-Hean Gan
TWI Ltd, Cambridge, UK
Paper presented at BINDT annual conference 2008, 15-18 September 2008, Macclesfield, Cheshire, UK.
Abstract
At a fixed frequency dispersion curves give fundamental informations on guided waves (wavelength, dispersion, phase and group velocities and the travelling time of each mode ...). In long range ultrasonic NDT inspection, the use of modelling can be used to predict the best choice of experimental conditions.
There is commercial software that produces dispersion curves (DC), e.g. Disperse®, however this software only works for cylindrical and plate structures. The solution of Disperse® is based on analytical solution of wave equation in the frequency domain. The Fourier Transform is used to convert the time equations in frequency domain and all variables are harmonic in infinite time.
The purpose of this project is to develop a Semi Analytical Finite Element Method (SAFEM) to obtain phase and group velocity dispersion curves, in order to generate ultrasonic guided waves for bar like structures with complex cross section.
The advantages of the SAFEM are:
- Deal with complex cross section structure.
- Fast calculation time.
- Small computational memory required compared with the ordinary finite element method.
In addition to dispersion curves, this method can be used to carry out a complete modelling of the modal propagation in complex cross section structures.
This work provides a powerful numerical model in the Non Destructive Testing (NDT) understanding. The numerical platform is then used to make expertise and provide database for several complicated situations such as I-beam inspection.
1. Introduction
Dispersion curves evaluation has been the subject of many researches, among the older research Nigro[1] in 1966 developed a semi-analytic solution based on function expansion of displacements called the Ritz method. Some difficulties were observed when dealing with convergence and the method was applied to square cross section bars. In the last 10 years analytical methods based on function expansion were not used because of their limitation to simple cross section and also because of their complexity. The finite element technique has been applied in all modelling fields in mechanics, especially in Long Range Ultrasonic (LRU) for the calculation of dispersion curves and mode characteristics. This method was also subjected to some limitation; mainly the memory used in such application is high because of the model length. SAFEM has been presented as an alternative as it avoids the 3D meshing and uses only 2D meshing. This is thanks to the analytical form of the considered displacements, the 3rd direction is simplified and 2D meshing is used for 3D displacement unknowns. In 1998 Volovoi[2] affirms that until that date dispersion curves were studied almost exclusively for rectangular or circular cross sections, while dispersion information for other important sectional geometries remains very scarce. He presented the SAFEM used to generate dispersion curves for non-homogeneous anisotropic I-beams. Gavric[3] used the SAFEM for LRU applications, he obtained the dispersion curves for propagative (real) and evanescent (complex) in a straight wave guide. The results are shown for a free rail for several deformations as axial, vertical flexion, horizontal flexion, torsional and other types of propagative waves which can not be described using analogies with simple beam waves. Gavric shows that polynomial approximation should be well chosen.
Taweel et al[4] used SAFEM for dispersion curves calculation of circular and rectangular cross section and also for three layers beam with anisotropic symmetry.
Recently many researches was published by Takahiro et al. In 2002 they presented the SAFEM applied to flexural mode focusing in pipes[5]. In[6] they used the SAFEM in advanced LRU modelling as a simulation and visualisation tool of wave propagation in plate and pipe with elbow and later in 2004[7,8] for square rod and rail. In [8], Takahiro et al present the group velocity and the mode deformation calculation based on modal expansion. This expansion uses the L and R respectively left and right eigenvectors for a given frequency.
There is an experimental method used to generate dispersion curves for complex cross section structures. This method applies the 2DFFT post-processing to time-space experimental data. Those data are obtained from different A-scan positions along the structure length. In [9] the excitation is a broad-band and the method is applied to generate dispersion curves for plate Lamb wave modes, with selective excitation for S0 and A0. The same method is used in [8] and [10] respectively applied to rail and plates, the used excitation is tone burst signal. In [11] Moilanen et al uses the 2DFFT to produce the phase velocity frequency spectrum identically to Takahiro in [8], Moilanen introduces a velocity filtering algorithm called selective 2DFFT that enhance the ability to discriminate wave modes. This technique is used to envelope the region of interest in the recorded signal and thus the undesired part of the time domain signal did not affect the spectrum. Moilanen et al applied their procedure in the dispersion curves generation in immersed plates.
In the present work, we present the SAFEM implementation and results for different cross section. The implementation will be limited to dispersion curve generation, the mode shape is obtained separately using standard software as the frequency and the wave-number are known. The validation of the SAFEM will be achieved with regard to published and to experimental results.
2. Semi analytical FEM
2.1 Governing equations
First, consider a prism element that consists of a small triangle on a cross-sectional xy plane and straight edges in the z-direction. The displacement and strain, at any point (x y z) in this element are written as:
λ).
[12]
ξ;
N3 = η represent the linear interpolation functions.
The elementary strain description is obtained by the gradient operator applied to the displacement vector:
(,d) represents the space derivative with respect to the direction (d). The inertia elementary term of (1) is:
The potential elementary term of the virtual work is:
The interpolated stiffness matrix over an element is expressed as
The potential elementary term is expressed as
In the absence of external forces, the virtual work is reduced to
