SIMULATION OF THE INSPECTION OF PLANAR NON MAGNETIC MATERIALS WITH ELECTRO MAGNETIC ACOUSTIC TRANSDUCERS Denis Prémel, C. Reboud, S. Chatillon, F. Reverdy and S. Mahaut CEA, LIST, Laboratoire Simulation et Modélisation, 91191 Gif-sur-Yvette CEDEX, France. ABSTRACT. For some specific applications in ultrasonic non destructive evaluation, EMATs (ElectroMagnetic Acoustic Transducers) are very useful for generating and receiving ultrasonic waves. EMAT works without any contact and liquid coupling. Various surface or bulk waves with any arbitrary polarities and orientations may be generated by changing the orientation of the magnets and the coils. Unfortunately, these types of probes show a poor sensitivity as receivers. CEA LIST has developed simulation tools, based on semi-analytical models dedicated to eddy current and ultrasonic testing, in order to predict signals obtained when inspecting planar structures. The first step of these developments concerns the inspection of conducting non-ferromagnetic materials. By combining eddy currents due to coils with the static magnetic field provided by magnets, the 3D Lorentz s force distribution is computed in the time domain and used as input for the semi-analytical ultrasonic models to compute the simulation of ultrasonic bulk waves and flaw interaction in the piece. This communication presents a specific configuration for our first experimental validation. The computation time is sufficiently low to perform parametric studies to improve the performances of the EMAT. Keywords: Simulation, EMAT Probes, Time domain, Eddy current NDE, Ultrasonic Bulk Waves, CIVA PACS: 43.38+n, 43.58+z, 07.64+z. INTRODUCTION The characterization or detection of flaws in ultrasonic inspection requires the generation of ultrasonic waves in solid materials. Conventional piezoelectric transducers are able to generate the ultrasonic energy but transfer from the transducer to the material under test requires the use of coupling. For some specific applications, the probe cannot be immersed or cannot be in contact with the specimen due to surface roughness or high temperature. For the inspection of conducting material, an alternative consists in applying an electromagnetic coupling. Electro-Magnetic Acoustic transducers (EMATs) have the capability to generate a great number of ultrasonic waves of different types without any contact or liquid coupling. Unfortunately, the poor efficiency of the transduction effect in the transmitter/receiver probe leads to a poor signal-to-noise ratio and the performances of this transducer must be absolutely optimized according to the application. This process often requires numerous parametric studies, so efficient simulation models are needed. CEA LIST aims in developing specific fast simulation tools, implemented in the CIVA platform, to help the end-user design such transducers according to the geometry of the specimen. In this paper, we are focused on the inspection of planar structures of non-ferromagnetic materials. Some semi-analytical ultrasonic models, dedicated to the propagation of ultrasonic waves in solid materials have been coupled to other several monochromatic semi-analytical eddy
current models in order to compute the response of an EMAT. These developments require to extend the monochromatic semi-analytical eddy current models to the time domain by using Fourier synthesis [1,2,3,4] to finally compute specific physical quantities, which constitute the inputs of ultrasonic models. This paper proposes to summarize the properties of each semi-analytical model, contributing to the global numerical model of an EMAT. A comparison between simulated data and experimental data shows the validity of the numerical model. Several imaging tools are also illustrated for the design of the transducer. PRINCIPLES OF ELECTROMAGNETIC ACOUSTIC TRANSDUCER Usually, an "EMAT" transducer consists of a set of current coils driven by a pulsed current inducing eddy currents within the skin depth of the specimen under test and a set of permanent or electromagnets to generate a static magnetic field. Figure 1.a depicts for example a simplified principle scheme. Let us consider a line current driven by a current varying in time. Figure 1.b shows an example of pulsed driving current. Pulsed eddy currents generated within the test specimen are in opposite to the exciting currents. The pulsed eddy current density is denoted by. A permanent magnet, located close to the coil, produces a static magnetic flux density, which is finally combined to eddy currents to generate body forces at the surface of the specimen. The Lorentz's force, given by = t (t), is mainly oriented, in the case illustrated in Figure 1, in the tangential direction. In most situations, for non-ferromagnetic materials, the dynamic magnetic field due to the pulsed current driving the exciting coil is assumed to be small compared to the external static magnetic field produced by the magnets. The Lorentz's force distribution due to the dynamic magnetic field is thus assumed to be negligible. Then, acoustic waves are propagating inside the specimen as guided (Lamb modes), surface or bulk waves modes according to the thickness of the specimen and the arrangement of the EMAT probe. Though an EMAT can generate a great number of waves (SH, LH, Rayleigh, Lamb) according to the configuration of the electromagnetic system, we are specifically interested in this paper by the generation of longitudinal waves, according to the configuration displayed in figure 2. Two rectangular current coils induce eddy currents in the slab according to the Y axis, the static magnetic field created by two permanent magnets is tangential to the surface of the slab, the Lorentz s force is normal to the surface and longitudinal UT waves are generated in the conducting slab. Figure 3 depicts the configuration, which is considered for modelling the EMAT, and the real transducer used to obtain experimental data for the validation of our model. Semi-analytical models dedicated to the computation, in the harmonic regime, of eddy currents induced in the slab were available into the CIVA platform. These models have been extended to work in the time domain by using Fourier Synthesis. The current density due to an arbitrary arrangement of several inductive coils may be calculated in the time domain in a 3D finite domain of the slab, taking into account arbitrary shapes for the driving current (see Figure 1.b). Semi-analytical ultrasonic models were already implemented for the computation of the propagation of ultrasonic waves into solid materials. The main development was to connect the electromagnetic model to the ultrasonic model, the former needs to provide an adequate source terms to the later. This way, the displacement field, the echo signal and the emitting/receiving probe configuration can be computed and displayed into CIVA. In the following parts of this paper, the two semi-analytical models are described and the connexion between them is explained. The experimental set-up, used to perform the
FIGURE 1 (a). A simplified scheme of an EMAT. FIGURE 1 (b). Example of driving current exciting the coils. FIGURE 2. Typical EMAT configuration, dedicated to the generation of longitudinal waves. experimental validation of the global model, is then presented. Finally, a short conclusion is proposed. THE SEMI-ANALYTICAL ELECTROMAGNETIC MODEL The purpose of this section is to present some details about the semi-analytical model dedicated to eddy current computations and an analytical model for the computation of the static magnetic field due to any arrangement of parallelepipedic permanent magnets. Computation of the static magnetic flux The computation of the 3D distribution of the static magnetic field produced by a set of two permanent magnets (see Figure 3) is first achieved, in the non magnetic case, by assuming the superposition of each 3D distribution obtained by each parallelepipedic permanent magnet. The analytical model is based on the scalar potential formulation and the magnetic charge viewpoint. The magnet is considered to be magnetized in one direction and saturated. Therefore, the magnetic vector M 0 is assumed to be constant and spatially uniform inside the magnet. The volumetric magnetic charge density ρ = M 0 is null and the magnetic field H can be deduced by an integral equation involving the fictitious surface magnetic charge density σ = M 0 n where n stands for the unit vector normal to the surface S of the rectangular magnet: =,. (1)
FIGURE 3. A schema of the configuration dedicated to the generation of longitudinal waves and a realization of the transducer which has been used for obtaining experimental data. Two rectangular coils are etched on a flexible Kapton film. The observation point and the source point are always distinct in our calculations., stands for the 3D Green's function. According to this formulation, it is possible to obtain some analytical expressions of the three components of the magnetic field for an arbitrary observation point in the free space out of the rectangular magnet. The numerical value of the magnetic vector M 0 is deduced from a practical value of the magnetic flux density measured at the center of a face of the magnet. Numerical results have been compared to those obtained by X. Gou [5] and another numerical 2D model developed at CEA LIST, which will be used to compute the static field inside ferromagnetic materials. Computation of eddy currents in the harmonic regime Any 3D geometrical configuration of the electromagnetic system may be interesting for the design of a new transducer. For such a 3D electromagnetic problem, the approach called Truncated Region Eigenfunctions Expansion (TREE) [6] developped at the University of Western Macedonia by T. Theodoulidis has been implemented into the CIVA platform [7] and various configurations of EMATs are now available in the new version of CIVA including notably pancake coils, linear coils, meander coils, as one can see in Figure 4. The principle of this approach consists in writing the 2D inverse Fourier transform of the eddy current density: =,, d d (2) where, stands for the expression of the generalized transmission coefficient due to a stratified medium. By choosing a finite box along directions and, so that the solution can be set to zero at its boundaries (and outside of the box), this equation can be translated into series instead of the integral corresponding to the infinite case. An analytical expression of the source term, may be derived for a great number of specific shapes of coils. When this term cannot be evaluated in a closed-form, it can be computed by using the 2D Fourier transform of each component of the magnetic flux density [6]:, = = = (3) Thus, it is possible to calculate the source term, for complex shapes of coils from numerical values of the normal component of the magnetic flux density, obtained in free space from the Biot-Savart law, at the interface of the piece. This step has been carried out by computing the magnetic field emitted by a set of current blocks of elementary shape.
FIGURE 4. Examples of complex shapes of exciting coils, which can be addressed in CIVA. Thus, any kind of complex shape may be represented by a finite number of trapezoids for which the analytical solution may be numerically obtained. The computational time is very low, in regard to the complexity of the shapes, which can be used in the new version of CIVA: D coils, rectangular coils with rounded corners or not, racetracks, meander coils and spiral coils. Calculation of the equivalent surface constraint in time domain Eddy currents are combined with the static magnetic field in a finite domain to evaluate the Lorentz s force:, =,,. For non-ferromagnetic materials, the body Lorentz's force due to the dynamic magnetic field is usually neglected. The time harmonic body force may be calculated at any observation point. However, the exciting coil is driven by a transient current and the body force must be evaluated in time domain. So, the temporal signal is evaluated by using Fourier synthesis. At each frequency, the body force is numerically evaluated in a finite domain limited by the skin depth depending on the exciting frequency. As the body force becomes localized near the surface, the zero th order moment of the force dominates the radiation and the surface stress is usually defined by using this first assumption [8]. Even if, in certain conditions, other moments must be considered [9], in a first step of our work, we derive the pulse surface equivalent constraint, by an integration with respect to depth. In time domain, we complete the formula:,, =,,, d d (4) Then, by assuming the separation of the spatial variables from the time variations, the two input parameters for the ultrasonic model are defined by the equivalent surface stress, and the temporal signal which is considered for ultrasonic time excitation:,, =, (5) THE SEMI-ANALYTICAL ULTRASONIC MODEL The semi-analytical model dedicated to the calculation of the radiation of ultrasonic waves in solid materials is based on the superposition of elementary Source points, which are distributed over the active surface of the probe. The surface of the transducer is first of all discretized by a set of source points. By using the pencil method applied to elastodynamics problems, the point spread function is determined taking into account the longitudinal and transversal waves, the reflection and mode conversions. Then, all contributions are integrated taking into account the distribution of the stress at the surface of the transducer. The global total displacement field is obtained by convolving the impulse response with the excitation signal. The main problem lies in the calculation of the point source Lamb s problem with a low computation time. Solving the point source Lamb s problem means calculating an exact or an approximated solution of the displacement field due to a punctual stress (normal or tangential) at the source point (Fig. 5).
(a) The source point problem FIGURE 5. The point source Lamb s problem. (b) The exciting signal in time domain Simulation tools, developed at CEA LIST, are based on a model of radiation from sources in direct contact with the component. In the simple case of an isotropic half-space, this model is equivalent to Miller and Pursey s model [11] for a source of normal stress and the Cherry s one [12] for a tangential stress. The impulse response, is obtained by a spatial convolution and the response is convolved with the exciting signal:, =,,, d d (6) The total displacement field is obtained by the convolution product, =, ). NUMERICAL RESULTS AND EXPERIMENTAL VALIDATION To validate the global semi-analytical model of the EMATS, a specific experiment has been performed. A NDT device, distributed by the M2M Company has been used to obtain the experimental data. The inspection configuration is defined in Figure 3. Two parallelepipedic permanent magnets create a tangential magnetic field in the region of interest. To illustrate the example chosen for this first experimental validation, Figure 6 (a),(b),(c) show respectively the eddy current distribution, the distribution of the magnetic flux density and finally the distribution of the Lorentz s force into the specimen, in the harmonic regime at a frequency of 100 khz. The first experimental validation, consist in the measurement of the displacement field generated by the EMAT and transmitted through an aluminum plate. The comparison between simulated and experimental cartographies is displayed in Figure 7 to 9: Figure 7 corresponds to a top view and Figure 8 corresponds to two slice views in the two scanning directions. These cartographies show a good agreement between experimental and simulated data. The prediction of the amplitude of the displacement field is very good as well as the dimensions of the focal spot. Figure 9 displays the waveforms for the A-Scan, some differences appear after the arrival of the longitudinal wave. They are currently under investigation. (a) (b) (c) FIGURE 6. Distribution of eddy currents (a), the magnetic flux density (b) and the Lorentz s force (c) into the specimen in the harmonic regime, at a frequency of 100 khz.
Experimental data Simulated data FIGURE 7. Cartographies of the experimental (left) and simulated (right) displacement fields. FIGURE 8. Experimental (left) and simulated (right) displacement fields for each slice. FIGURE 9. Experimental (top) and simulated (bottom) Ascan observed at the center of the probe. CONCLUSIONS AND FUTURE WORKS A semi-analytical model dedicated to the simulation of the inspection of a planar structure by an EMAT has been developed. The time computation is sufficiently low to perform parametric studies to optimize the performances of an EMAT. Complex shapes of coils and arrangements of parallelepipedic permanent magnets may be simulated in order to build up any complex 3D electromagnetic system. The numerical model takes into account any shape of the pulsed driving current in the exciting coil and some physical quantities such as the displacement field into the specimen and the echo signal provided by the emitting/receiving EMAT can be displayed by CIVA. Bulk waves generated by the transducer can interact with the backwall of the planar structure or with a flaw located in the slab. Each contribution due to longitudinal and transversal modes can be displayed, either separately or both together. Some experimental data show the
validity of our first numerical model into specific favorable conditions. Other comparisons with experiments will be performed to complete the experimental validation. This work opens up several perspectives. First, the case of planar structures made of ferromagnetic materials will be investigated, by taking into account magnetostrictive effects. Other modules implemented in the CIVA platform will be used in order to simulate guided modes generated by EMATs in planar structures. Finally, some additional developments will be achieved to simulate inspections with phased arrays made of EMAT elements. ACKNOWLEDGEMENTS This work is supported by the TECNA research program for sodium cooled reactors. REFERENCES 1. C.C. Tai, J.H. Rose, and J.C. Moulder, Review of Scientific Instruments, 67 (11), pp.3965-3972 (1996). 2. H.C. Yang and C.C. Tai, Measurement Science and Technology, 13, pp.1259-1265, (2002). 3. Y. Li, G.Y. Tian and A. Simm, NDT & E International, 41(6), pp.477-483, (2008). 4. J. Pavo, IEEE Transactions on Magnetics, 38(2), pp.1169 1172,(2002). 5. X. Gou, Y. Yang, and X. Zheng, Journal of Applied Mathematics and Mechanics, 25(3), pp. 297-306, (2004). 6. T. P. Theodoulidis and E. E. Kriezis, Eddy current canonical problems (with applications to nondestructive evaluation), Tech Science Press, (2006). 7. C. Reboud and T. Theodoulidis, 16 th International workshop on Electromagnetic Nondestructuctive Evaluation (ENDE 2011), IIT Madras, Chennai, India, (2011) 8. K. Kawashima, IEEE transactions on sonics and ultrasonics, 31(2), pp. 83-94, (1984). 9. R. Bruce Thompson, Journal of Nondestructive Evaluation, 1, pp.79-85, (1980). 10. www.civa.cea.fr. 11. G. F. Miller and H. Pursey, Proc. R. Soc. Lond. A223, pp.521-541, (1954). 12. J.T. Cherry, Bull. Seismol. Soc. Am. 52, pp.27-36, (1962).