Abdellahi Abderahmane

Bastien Clausse

Alain Lhémery
email: alain.lhemery@cea.fr

Laurent Daniel

Mechanisms of elastic wave generation by EMAT in ferromagnetic media

Keywords:

Introduction

In ultrasonic non-destructive techniques, piezoelectric transducers, laser-ultrasonic systems, and electromagnetoacoustic transducers (EMAT) are the most widespread means for generating elastic waves in metallic structures. The piezoelectric techniques use piezoelectric ceramic transducers bound to the structure or radiating from a coupling medium (either solid or fluid). These techniques' need for mechanical contact impedes their potential use in numerous applications. Although optical techniques (using a laser source) do not suffer from this drawback, they are difficult to implement, requiring the surface condition of the part to be inspected, which is sometimes electromagnetic (EM) source) EMAT can generate a wave in ferromagnetic (FM) materials through three mechanisms: a) Lorentz force (Lf), b) magnetic force, and c) magnetostriction strain. While the first two are indeed forces, the last one is an eigenstrain (like thermal expansion) caused by the magnetic field and can -for convenience -be represented by a fictitious equivalent force term. Though the theory of wave generation by EMAT in nonmagnetic conductive materials -in which only Lf is created -is rather simple and well established (Gaerttner et. al. [11]), that, for the case of ferromagnetic (FM) materials is far more complex to establish. Such materials are widely used across various industries. Their ultrasonic characterization -like that of nonmagnetic materials -often requires the transducer to select specific types of waves, which can be done using EMATs (see [START_REF] Thompson | Microstructure-861 independent acoustoelastic measurement of stress[END_REF][START_REF] Clark | Residual stress determination 864 in aluminum using electromagnetic acoustic transducers[END_REF][START_REF] Hirao | Resonance EMAT system 867 for acoustoelastic stress measurement in sheet metals[END_REF][START_REF] Kato | Spatial localization 870 of stress-perturbing wave generated by an Electromagnetic 871 Acoustic Transducer[END_REF][START_REF] Pei | A new method for plastic 874 strain measurement with Rayleigh wave polarization[END_REF][START_REF] Liu | 877 Development of a directional magnetic-concentrator-type 878 electromagnetic acoustic transducer for ultrasonic guided 879 wave inspection[END_REF][START_REF] Zhang | Modes control of Lamb wave 881 in plates using meander-line Electromagnetic Acoustic 882 Transducers[END_REF][START_REF] Thompson | Generation of horizontally polarized shear 884 waves in ferromagnetic materials using magnetostrictively 885 coupled meander-coil electromagnetic transducers[END_REF][START_REF] He | A New Electromagnetic 888 Acoustic Transducer design for generating and receiving S0 889 Lamb waves in ferromagnetic steel plate[END_REF][START_REF] Thompson | A model for the electromagnetic 892 generation and detection of Rayleigh and Lamb waves[END_REF]). The fact that wave generation mechanisms for ferromagnetic materials can interact constructively or destructively hinders the ability to select or even generate elastic waves. Several attempts combining distinct theoretical models have been proposed to solve the EMAT wave generation problem in ferromagnetic materials, which led to some contentions in the recent ultrasonic literature. To understand why, it is recalled that various methods exist to model the EM force (the combination of Lorentz and magnetic forces) exerted on a FM object in an EM field. For instance, the equivalent magnetic sources method (see [START_REF] Vandevelde | A Survey of magnetic 897 force distributions based on different magnetization models 898 and on the virtual work principle[END_REF][START_REF] Sanchez-Grandia | General formulation for magnetic forces in linear 902 materials and permanent magnets[END_REF]), formulates the problem in terms of the interactions of the EM field with equivalent magnetic currents and charges, Kelvin's method (see [START_REF] Melcher | Continuum Electromechanics[END_REF]), in terms of the interactions with magnetic moments, Korteweg-Helmholtz (see [START_REF] Melcher | Continuum Electromechanics[END_REF][START_REF] Barré | Experimental validation 907 of magnetic and electric local force formulations associated 908 to energy principle[END_REF]) and Maxwell stress tensor (Max-ST) methods (see [START_REF] Sanchez-Grandia | Magnetostatic Maxwell's tensors in magnetic media 911 applying virtual works method from either energy or co-912 energy[END_REF][START_REF] Bossavit | Bulk forces and interface forces in assemblies of magnetized pieces of matter[END_REF]) both use the virtual work principle applied to the EM energy. All four formulations give the same global force exerted on the object (see [START_REF] Grandía | General formulation for magnetic forces in linear materials and permanent magnets[END_REF]) as the sum of volume and surface terms. However, they yield different force distributions. As a result, some bad practices arose, in which the volume term from one method is combined with the surface term from another method. Such operation in ultrasonic applications -where the generated wave type is governed largely by the distribution of force -led, unescapably, to consequential discrepancies. A detailed discussion on the hazardous nature of this approach is provided by Seher and Nagy [START_REF] Seher | On the separation of Lorentz and magnetization forces in the transduction mechanism of Electromagnetic Acoustic Transducers (EMATs)[END_REF]. In addition to the theoretical intricacies, practical questions regarding the dominant wave generation mechanism (WGM) were raised. Magnetostriction was the only mechanism considered by Ribichini et al. [START_REF] Ribichini | Quantitative modeling of the transduction of Electromagnetic Acoustic Transducers operating on ferromagnetic media[END_REF], in which the EMAT static and dynamic magnetic fields were perpendicular. Such EMAT was used to generate SH0 in a nickel plate. Another EMAT, for which the static and dynamic magnetic fields were parallel (see [START_REF] Ribichini | Study and comparison of different EMAT configurations for SH wave inspection[END_REF][START_REF] Hirao | An SH-wave EMAT technique for gas pipeline inspection[END_REF]) was studied. Only Lorentz force (Lf) was considered when such EMAT was used to generate SH0 in steel. In [START_REF] Ribichini | Quantitative modeling of the transduction of Electromagnetic Acoustic Transducers operating on ferromagnetic media[END_REF][START_REF] Ribichini | Study and comparison of different EMAT configurations for SH wave inspection[END_REF] the EMAT coil was a single current-carrying wire, and in [START_REF] Hirao | An SH-wave EMAT technique for gas pipeline inspection[END_REF] a rectangular spiral coil. An EMAT with a meander coil was studied by Thompson [START_REF] Thompson | Mechanisms of electromagnetic generation and detection of ultrasonic Lamb waves in iron nickel alloy polycrystals[END_REF] where, in contrast with previously cited works with parallel static and magnetic 125 fields, Lf was not neglected by the model and was shown 126 to be dominant in comparison to magnetostriction for 127 large static magnetic field. This was observed for three 128 different FM materials. For low static magnetic field, 129 magnetostriction was the dominant mechanism. Whilst 130 agreement between the model and experiment was mostly 131 good, the latter showed tendencies that the model did not 132 predict, and the author suggested a third WGM, that later 133 on [START_REF] Thompson | Physical principles of measurements with EMAT transducers[END_REF] was shown to be the magnetic force. Ashigwuike et 134 al. [START_REF] Ashigwuike | A study of the transduction mechanisms of electromagnetic acoustic transducers (EMATs) on pipe steel materials[END_REF] one identifies the parameters of both models are given in 295 [START_REF] Rouge | Transformation of body force localized near the surface of a half-space into equivalent surface stresses[END_REF] and [START_REF] Li | Stress effects on ferromagnetic materials: investigation of stainless steel and nickel[END_REF], respectively. Expressions of magnetization 296 and magnetostriction strain as functions of 𝑯 and material 297 parameters are not reported here due to lack of space and 298 are readily found in [START_REF] Daniel | A simplified 3D constitutive law for magneto-mechanical behaviour[END_REF][START_REF] Jiles | Theory of the magnetomechanical effect[END_REF] force and the volume magnetic force) the equivalent 308 magnetostriction force 𝒇 𝑚𝑠 and the surface magnetic 309 traction 𝒕 𝑚 . All these forces can be represented using the 310 augmented Maxwell tensor given below (see Bossavit [START_REF] Bossavit | Bulk forces and interface forces in assemblies of magnetized pieces of matter[END_REF] for 311 detailed derivation) under the assumption that magnetic 312 induction (𝑩 = 𝜇 0 (𝑴 + 𝑯)) and field (𝑯) are collinear: 313

𝝈 𝑀𝑥 = 𝑩 ⊗ 𝑯 + 𝐺(𝑯)𝑰 -𝑪: 𝝐 𝑚𝑠 (1)
where 𝑯 obeys the following Maxwell's equations in 314 matter given by ∇ × 𝑯 = 𝑱 𝑒 and ∇ ⋅ 𝑯 = -∇ ⋅ 𝑴. The 315 second order tensor 𝑩 ⊗ 𝑯 is given by (𝑩 ⊗ 𝑯) 𝑖𝑗 = 𝐵 𝑖 𝐻 𝑗 316 and 𝑪 is the elastic stiffness tensor. Finally, 𝐺(𝑯) denotes 317 the Gibbs energy, defined as (see Henrotte et al. [START_REF] Henrotte | Electromagnetic force density in a ferromagnetic material[END_REF]): 318

𝐺(𝑯) = -∫ 𝐵(𝒉) ⋅ 𝑑𝒉 |𝑯| (2)
In EMAT applications, the magnetic field and, 319 consequently, all quantities dependent on it 320 (magnetization and magnetostriction) can be decomposed 321 into two parts: a static part stemming from the permanent 322 magnet(s) and a dynamic part from the coil(s). In general, 323

|𝑯 𝑑 | ≪ |𝑯 𝑠 | (i.e., the hypothesis of piezomagnetism 324 (PMH)). Under this hypothesis, one can write (using 325

Taylor series expansion to the first order): 326

𝑩(𝑯 𝑠 + 𝑯 𝑑 ) = 𝑩 𝑠 + [ 𝜕𝑩 𝜕𝑯 ] 𝑯 𝑠 ⋅ 𝑯 𝑑 (3)
𝝐 𝑚𝑠 (𝑯 𝑠 + 𝑯 𝑑 ) = 𝝐 𝑠 𝑚𝑠 + [ 𝜕𝝐 𝑚𝑠 𝜕𝑯 ] 𝑯 𝑠 ⋅ 𝑯 𝑑 (4)
𝐺(𝑯 𝑠 + 𝑯 𝑑 ) = 𝐺 𝑠 + [( 𝜕𝐺 𝜕𝑯 )] 𝑯 𝑠 ⋅ 𝑯 𝑑 (5)
where in Eq. ( 3) (resp Eq. ( 4)) the quantity in brackets is 327 the static differential magnetic permeability 𝝁 𝑠 (resp the 328 magnetostrictive coupling tensor 𝒅 𝑚𝑠 ). Recalling that 329

𝑩 𝑠 ∥ 𝑯 𝑠 , one can readily show that [ 𝜕𝐺 𝜕𝑯 ⁄ ]

𝑯 𝑠 = 𝑩 𝑠 330 Substituting Eq. ( 3), ( 4) and (5) into Eq. ( 1) yields three 331 terms: (a) a static term (b) a term that depends on the 332 product of 𝑩 𝑠 ⋅ 𝑯 𝑑 and (c) a term of higher orders in 𝑯 𝑑 . 333

Owing to its nature, the static term does not generate 334 waves. The one with higher orders of 𝑯 𝑑 is disregarded 335 due to its relatively small amplitude (under PMH), so only 336 the second term is of interest in the present work. The 337 divergence of this term gives 𝒇 𝑒𝑚 + 𝒇 𝑚𝑠 and the jump of 338 its magnetic part (i.e., without -𝑪: 𝝐 𝑚𝑠 ) across the interface 339 (𝛤) of two media with different magnetic permeabilities 340 gives 𝒕 𝑚 . These expressions are given by: 341 burden. Details of the method allowing such conversion 357 are provided in [START_REF] Rouge | Transformation of body force localized near the surface of a half-space into equivalent surface stresses[END_REF] for planar and in [START_REF] Clausse | Transformation of body force generated by noncontact sources of ultrasound in an isotropic solid of complex shape into equivalent surface stresses[END_REF] for arbitrary 358 surfaces. The method assumes the volume force 359 distribution to be confined near the surface so that the 360 elastic wavelength should be larger than the force 361 distribution dimension in the piece thickness. Such an 362 assumption is almost always valid in the case of EMAT in 363 ferromagnetic media since the force distribution dimension 364 in the thickness is governed by 𝑯 𝑑 which does not 365 penetrate such media more than a few skin depths. Given 366 typical frequencies in EMAT applications, the magnetic 367 permeability, and elastic properties of ferromagnetic 368 media, the elastic wavelength is at least two orders of 369 magnitude larger than the skin depth. By using a Taylor 370 series expansion to the second order of the radiation 371 integral of a body force distribution (given in Eq.9), the 372 method [START_REF] Clausse | Transformation of body force generated by noncontact sources of ultrasound in an isotropic solid of complex shape into equivalent surface stresses[END_REF] makes it possible to rewrite the volume 373 integrals in the form of a surface integral over an 374 equivalent stress distribution (Eq.10). The full derivation 375 [START_REF] Clausse | Transformation of body force generated by noncontact sources of ultrasound in an isotropic solid of complex shape into equivalent surface stresses[END_REF][START_REF] Rouge | Transformation of body force localized near the surface of a half-space into equivalent surface stresses[END_REF] was proposed under the assumption of an 376 elastically isotropic material, but the anisotropic case can 377 be treated the same way (see [START_REF] Clausse | Modélisation des traducteurs 976 électromagnétiques acoustiques (EMAT) pour le contrôle 977 non-destructif (CND) de milieux ferromagnétiques[END_REF]). We have: 378 Once 𝝈 𝑡𝑜𝑡 (i.e., the wave source) has been obtained, the 394 elastodynamic wave radiation problem written as 395 given in Table 1. All measurements are made by moving 477 the 2-turn coil along the EMAT coil diameter and 478 maintaining its distance from the test piece at . 479 𝐵 𝑑 (𝑡) signal (not shown) is similar to that of current 480 (Fig. 3a). ), the sensor is at 𝑧 = 1.5 𝑚𝑚 of the steel (resp. aluminum) sample surface. 497

𝒇 𝑒𝑚 = ∇ ⋅ (𝑩 𝑠 ⊗ 𝑯 𝑑 + 𝝁 𝑠 𝑯 𝑑 ⊗ 𝑯 𝑠 -(𝑩 𝑠 ⋅ 𝑯 𝑑 )𝑰) (6) 𝒇 𝒎𝒔 = -∇ ⋅ (𝑪: (𝒅 𝑚𝑠 ⋅ 𝑯 𝑑 )) (7) 𝒕 𝑚 = [𝑩 𝑠 ⊗ 𝑯 𝑑 + 𝝁 𝑠 𝑯 𝑑 ⊗ 𝑯 𝑠 -(𝑩 𝑠 ⋅ 𝑯 𝑑 )𝑰] 𝛤 (
∀𝒙 ∈ Ω,
(Ω) ∇ ⋅ 𝝈 = 𝜌 𝜕 2 𝒖 𝜕𝑡 2 (11)
(∂Ω δ ) 𝝈 ⋅ 𝒏 = 𝝈 𝑡𝑜𝑡 (12

Piezomagnetic hypothesis (PMH)

EMAT induces static (due to the magnet) and dynamic (due to the coil) magnetic fields in FM materials. The PMH assumes the latter to be sufficiently smaller than the former so that the dynamic problem can be treated as a perturbation of the static one. To validate the PMH, we take the example of a test piece (a cylinder of 70 𝑚𝑚 diameter and 147 𝑚𝑚 height) made of Z20C13 stainless steel. Measured anhysteretic magnetization and magnetostriction curves are given in Fig. 4. To avoid issues stemming from EMAT wave detection and to concentrate on wave generation, a focused piezoelectric transducer (PZT) from Olympus Panametrics of 5.5cm focal length and 2MHz central frequency is used for detection. A calibration experiment whose results are not shown helped identify the current range for which the EMAT generates detectable signals. Although sensitive only to the normal displacement component, the focused PZT can detect shear waves due to their oblique incidence. In other words, due to the size of the sample, the S-wavefront is still spherical when it reaches the bottom surface. It has, therefore a normal component which can be detected by the sensor. The experiment is schematized in Fig. 5. The EMAT, the test piece and the PZT are of cylindrical shape and are axisymmetrical relatively to the same axis. The distance between the PZT and the bottom surface of the test piece is maintained at 50mm (equal to the PZT focal distance). The lower half of the test piece and the PZT are immersed in a water tank. A Rogowski coil is used to measure the current (5 cycles of 1 MHz) from the signal generator to the EMAT coil. The lift-off is maintained at 4 𝑚𝑚. Such configuration generates both longitudinal (L) and shear (S) bulk waves in FM materials. To further consolidate this, the maximum amplitudes of measured signals for both L and S waves as functions of are given in Fig. 6 (e) and (f). One notices a linear relationship between wave amplitudes and . The latter is proportional to (Maxwell-Ampère) in consequence, a similar linear relationship must also exist between 𝐻 𝑑 and wave generation mechanisms, which is exactly what PMH predicts (see Eq.6-8). Finally Fig. 6g gives the time signal of the excitation current and measured voltages by the PZT sensor. The observed DC offset in current is due to the excitation systems, it doesn't however effect the validity of the hypothesis. In conclusion, these results suggest that higher orders of 𝐻 𝑑 that would lead to a nonlinear relationship between current and elastic wave amplitudes and would distort the time signal are negligible, thereby consolidating the validity of PMH.

L and S generation and radiation in FM materials

Now that both the ISH and PMH are verified, one can study the ultrasonic field associated with bulk waves of both polarities radiated by EMAT. For this, the experimental setup is the same as in the previous subsection, with the difference that here 𝐼 𝑒 is maintained at 2.42 A and the PZT is moved to scan the bottom surface of the test piece. Regarding the simulation, both nonlinear (given in Fig. 4a) and linear (given by the slope at the origin of curve in Fig. 4a) magnetic constitutive laws were used. Measured (resp simulated) C-scans (maximum amplitude of the wave displacement field at each scanning position) for both L and S waves are presented in Fig. 7 (a) and (b) (resp Fig. 7 (c) and (d)). A comparison of simulated and measured results along the test piece diameter is given in Fig. 7 (e) and (f). All results are normalized using their respective S wave amplitude at the center of the test piece. The dissymmetry in experimental measurements comes 579 from that of the coil. The return wire starts from the coil 580 center and crosses its section: this causes it to tilt (see 581 Fig. 5), thereby breaking the symmetry. This is evidenced 582 by the slight shift (to the right) of the central blue spot. 583

Secondary lobes around ±27 𝑚𝑚 are also present in the 584 measurements. These are not predicted by simulation 585 using linear magnetic constitutive laws but are predicted 586 when using nonlinear laws. Moreover, the former 587 simulation underestimates the L wave amplitudes along 588 the diameter, contrary to the latter. and surface), and magnetostriction strain. Whilst this is 594 well established in the literature (Thompson [24]), no 595 work, to the best of our knowledge, has considered all 596 three mechanisms together, and no studies were carried 597 out to compare these mechanisms to each other, in 598 different configurations. Such comparison offers important 599 information that should be considered in designing EMAT 600 and interpreting measurements made in ferromagnetic 601 media. 602 Due to the prohibitive number of inputs in an EMAT 603 experiment (shapes of the coil(s) and magnet(s), current 604 intensity and frequency, lift-off, material properties, …), it is extremely difficult to rely solely on the experiment to carry out parametric studies. Conversely, simulation offers both the speed and flexibility required to do so. The present section is divided as follows: In Sec. 4.1 WGMs and their corresponding radiated waves are studied as functions of material properties. In Sec. 4.2 they are studied as functions of EMAT excitation (i.e., the current in the coil and the static magnetic field of the magnet).

WGMs as functions of material properties

Four materials (nickel, Z20C13, AISI410, and low carbon steel (LCS)) of similar elastic but contrasting electromagnetic properties are studied. Magnetic constitutive laws (𝑴(𝑯)) are assumed linear. This assumption -not the most suitable to accurately describe experimental data in some cases (see Fig. 7) -allows for consequential computational simplification, which is needed to carry out parametric studies. Moreover, nonlinearity is not required to prove the premise of the paper (i.e., all WGMs should be included when designing an EMAT). In both subsections, magnetostrictive constitutive laws are obtained from various approaches. SMSM [START_REF] Daniel | A simplified 3D constitutive law for magneto-mechanical behaviour[END_REF] was used for the anhysteretic curve of nickel. It is measured for that of Z20C13 [START_REF] Rouge | Modélisation du rayonnement ultrasonore par 980 un traducteur EMAT dans une pièce ferromagnétique[END_REF]. The phenomenological approach [START_REF] Jiles | Theory of the magnetomechanical effect[END_REF] was used to that of AISI410. Finally, the curve of LCS was obtained by interpolating the corresponding curve appearing in Hirao and Ogi [START_REF] Hirao | EMATs for science and industry: 983 noncontacting ultrasonic measurements[END_REF]. These curves are shown in Fig. We note that it is nonetheless a mistake to consider only 665 such force in this case since the magnetic bulk force, 666 however small it may be, generates longitudinal waves 667 that are almost not generated by tangential Lf (only by 668 "edge diffraction" which would be nonnegligible only if the 669 source distribution showed a sharp discontinuity 670 (Lhémery [39])). 671

672

This point is discussed in more details in Sec.4.1.2. 673

Consider now the case of nickel (see Fig. 9). This material 674 has the strongest electric, magnetic, and magnetostrictive 675 properties amongst the four studied materials. Expectedly 676 -from the EMAT geometry -tangential components of magnetic traction and magnetostriction are small compared to the electromagnetic force. Looking at 𝝈 𝑒𝑚 , one can see that the contribution of bulk magnetic force (which is primarily represented by 𝜎 𝑧 𝑒𝑚 ) is small compared to that of Lorentz force (represented by 𝜎 𝑥,𝑦 𝑒𝑚 ). At first glance, this seems in contradiction to the fact that this material has the highest magnetic permeability. Actually, high permeability means, on the one hand, a high magnetic traction (as seen in the figure) but also a small skin depth, which in turn reduces the volume in which the bulk magnetic force operates. The small skin depth also affects Lorentz's force, which, in the present case, is slightly smaller than the equivalent force due to magnetostriction. The large value of the latter force is attributed to its relatively large magnetostriction strain (Fig. 8). The example of nickel shows a case in which Lorentz force is no longer the dominant mechanism, and in contrast to the previous case, the normal component of the wave source (𝜎 𝑧 𝑡𝑜𝑡 ) is mainly due to magnetostriction and magnetic traction (instead of the magnetic bulk force for LCS).

EMAT geometry and previous results (for LCS and nickel) suggest that magnetic and magnetostrictive properties manifest themselves mostly through the normal component of WGMs. On this account, for the last two materials (AISI 410 and Z20C13), only normal components are shown (in Fig. 9), tangential components being dominated by Lorentz force. Unsurprisingly, the magnetic bulk force for both materials is relatively small due to the relatively higher permeability. As for the contribution to the normal component of the wave source (𝝈 𝑡𝑜𝑡 ), one sees that it is dominated by magnetic traction for Z20C13 (due to its relatively weak magnetostriction) and is almost evenly distributed between magnetostriction and magnetic traction for AISI410.

The unexpected negative magnetic traction stems from the dynamic problem (wave generation) being treated as a perturbation of the static one (with a given static magnetic traction). Therefore, when 𝑯 𝑑 opposes 𝑯 𝑠 , the material is less pulled towards the magnet, which translates into a traction smaller than that in the static (reference) problem, leading to a negative difference. This also applies to magnetostriction strain. Finally, the observed irregularity in the rotundity of the z-components is caused by the non-axisymmmetric shape of the spiral coil.

In conclusion, an EMAT with a fixed excitation and geometry, would yield in different materials, WGMs with contrasting magnitudes. A dominant WGM in one material might be negligible in another one. Furthermore, due to the vectorial nature of WGMs and the fact that each elastic wave has a specific particle displacement, each WGM vectorial component needs to be studied separately. This point is treated in more detail in the next subsection.

751

To understand why this is so, one recalls that the latter is 752 generated mostly by a single WGM (Lorentz force), while 753 the former is generated by both magnetic force (bulk and 754 surface) and magnetostriction. Such WGMs do not always 755 have the same spatial distribution and can interact 756 destructively, depending on the material magnetostriction 757 properties. 758

817

Present results showed that assuming linear magnetic 818 behavior is sufficient to prove the premise of the paper 819 (i.e., wave generation mechanisms depended strongly on 820 material properties and on EMAT excitation). However, 821

this assumption may lead to inaccuracy, as seen in Fig.

Fig. 1 . 3 . 446 Fig. 2 .

 134462 Fig. 1. Schematic representation of the four-step solution of the problem of EMAT generation of elastic wave in ferromagnetic materials. Where the inputs, ouputs methods and models of each step alongside their corresponding references are recalled. 1: Computing induced eddy currents, magnetization, and magnetostriction strain. 2: Computing bulk forces distributions. 3: Converting bulk force distributions to surface force distribution. 4: Computing radiated elastic waves.

3. 1 .

 1 Infinitesimal strain hypothesis (ISH) 447 The EMAT induces both static and dynamic phenomena 448 in FM materials. The former stems from the magnet that 449 exerts a magnetic force and induces a magnetostriction 450 strain in the material. The latter results from the EM fields 451 radiated by the coil. Their magnitudes depend on material 452 properties (elastic, magnetic, and magnetostrictive) and 453 on electrical excitation. The ISH (needed in the 454 development of the solution for the wave generation 455 problem) assumes such deformations to be sufficiently 456 small for the computation of EM fields on the undeformed 457 shape -of the test piece -to give accurate results. 458 Because induced EM fields in materials are not measured 459 directly, we compare measured and computed (by CIVA-460 ET) induced electromotive forces (EMF) to validate ISH. Two 461 configurations are studied, in both, the experimental setup 462 is identical to that in Fig.5 (without the water tank and 463 PZT sensor). In the first (resp. the second) the EMAT is 464 placed above a steel (resp. aluminum) test piece at a 465 distance of 4 𝑚𝑚. In both cases, the current in the coil is 466 a pulse of 2 MHz center frequency shown on Fig.3a. A 467 small 2-turn coil (of section ) is used to measure 468 the induced EMF, from which the electromagnetic 469 induction 𝑩 𝑑 is deduced. This coil is placed between the 470 EMAT and the test piece and can be moved horizontally 471 (i.e., parallel to the coil plane). A Rogowski coil is also used 472 to measure the current in the EMAT coil, this signal being 473 used in CIVA-ET to compute EM fields. Results from this 474 experiment are given in Fig.3.b (for steel) and Fig.3.c (for 475 aluminum). The material properties of each test piece are 476

  Figures 3.b and 3.c show the results for 481 𝑚𝑎𝑥 ⏟ 𝑡 |𝐵 𝑑 (𝑡)| at each position. 482

Fig. 3 .

 3 Fig.3. (a) current circulating in the EMAT coil. (b) (resp (c)) comparison of measured and simulated EM induction for different spatial positions along the diameter of the coil (centered at), the sensor is at 𝑧 = 1.5 𝑚𝑚 of the steel (resp. aluminum) sample surface.

Fig. 4 .Fig. 5 .

 45 Fig.4. Measured magnetization (a), and magnetostriction (b) as functions of the magnetic field for Z20C13.To reduce noise, measured signals were averaged over 256 shots. Throughout the present document, longitudinal and shear waves were separated experimentally using their time-of-flight while the simulation tool for field computation (CIVA) is semi-analytical and modal, thus, gives access to their respective fields separately. It is worth noting that in order to compare the signal shapes, all signals were normalized, and measured signals were timeshifted. This allowed superimposing simulated and

Fig. 6 .

 6 Fig.6. (a) and (b): comparison of measured (blue marker) and simulated (red solid line) time signals for both L and S bulk waves for I = 2.42A. (c) and (d): the Fourier transform of the time signals. (e) and (f): the maximum (measured) amplitude of L and S waves as function of current. (g): measured electric current in the excitation coil and the voltage in the PZT sensor.

Fig. 7 .

 7 Fig.7. (a) and (b) (resp (c) and (d)): experimental (resp simulated) C-scans for L-wave and S-wave. (e) and (f): comparison of measured (blue markers) and simulated field amplitudes using nonlinear (solid red line) and linear (dotted red line) magnetic constitutive laws.

  589

4 .

 4 Parametric studies on EMAT wave generation 590 mechanisms 591 As stated in Sec.2 EMAT generates waves through three 592 mechanisms: Lorentz force, magnetic force (both volume 593

8 .

 8 Throughout this subsection, current intensity and frequency are 𝐼 𝑒 = 0.1 A, and 𝑓 = 1 MHz, and the normal magnetic induction of the magnet at its center is 𝐵 𝑧 = 1.5 T. Sec.4.1.1 treats WGMs and Sec.4.1.2 their corresponding radiated waves.

Fig. 9 .

 9 Fig.9. The three wave generation mechanisms for the four studied materials. (For each material) First column: electromagnetic force, second: magnetostriction equivalent force, third: magnetic traction and in the last: total force distribution. For a comprehensive illustration, each component is normalized using max(max(𝜎 𝑥 𝑡𝑜𝑡 ) , max(𝜎 𝑦 𝑡𝑜𝑡 ) , max(𝜎 𝑧 𝑡𝑜𝑡 )).

4. 1 . 2 .

 12 Radiated elastic waves 732 Previous wave sources 𝝈 𝑡𝑜𝑡 (for each material) are now 733 used by CIVA-UT to compute the radiated elastic field. In 734 this simulation, the test piece is assumed to be semi-735 infinite and wave amplitudes are taken below the center 736 of the EMAT at a depth of 𝑧 = 75 mm. The time-737 dependent z-displacement waveforms at this point are 738 shown in Fig.10. The first wave packet (with the shortest 739 time of flight ∈ [13, 14]𝜇𝑠) corresponds to the fastest 740 wave, i.e., the longitudinal wave (L). As expected, in LCS 741 the dominant Lorentz force (mostly tangential) gives rise 742 to a shear (S) wave amplitude larger than that of the 743 longitudinal L-wave. In nickel, the dominant 744 magnetostriction equivalent force (mostly normal) gives 745 rise to an L-wave amplitude larger than that of the S-746 wave. For the two intermediate materials (AISI410 and 747 Z20C13) both L and S waves are generated with similar 748 magnitudes. Interestingly the L wave time signature seems 749 to depend on the material, unlike that of the S wave. 750

Fig. 10 .

 10 Fig.10. Time-dependent particle displacement L and S waves radiated in the four studied materials. The first packet (with the shortest time of flight ∈ [13, 14]𝜇𝑠) corresponds to L wave.

Fig. 12 .

 12 Fig.12. Maximum amplitudes for each WGM for three different materials (LCS, Ni, and Z20C13), as functions of current intensity (a), and static magnetic induction (c) LCS, (d) Ni and (d) Z20C13.

The theoretical model of EMAT generation in 200 ferromagnetic media

  : a dimensionless parameter representing the 290 interdomain coupling (𝛼), 𝑀 𝑠 , a parameter characterizing 291 the shape of the anhysteretic magnetization (𝑎), two 292 parameters fitted on the anhysteretic magnetostriction 293 curve (𝛾 11 and 𝛾 21 ). Experimental procedures from which 294

	181	convolution of the appropriate Green's tensor and	235	Elastic wave generation in FM materials by EMAT is a
	182	equivalent surface stress distributions. Sec. 3 is dedicated	236	multiphysics problem. To solve it, we divided it into four
	183	to experimental studies to verify the validity of ISH	237	sub-problems, each of them specific to a given physics.
	184	(Sec.3.1) and PMH (Sec. 3.2). In Sec.3.3, measured and	238	Methods of solutions to these problems have already been
	185	simulated radiated field distributions of both longitudinal	239	developed -separately -in the literature, sometimes by
	186	(L) and transverse (T) waves are compared to validate the	240	one of the authors of the present paper. They are
	187	overall modelling approach. Finally, in Sec.4, parametric	241	combined and ordered in the present work following four
	188	studies are carried out to compare all WGMs generated	242	steps (see Fig.1) to reach the solution to the complete
	189	by a given EMAT in four different FM materials. Each	243	wave generation problem (WGP). For conciseness, they
	190	component of each transduction mechanism is treated	244	are only briefly reported hereafter, with references to their
	191	separately to ease their comparison. In Sec. 4.1 WGMs	135 245	considered the two WGMs previously studied but detailed derivations being provided.
	192 193	and their corresponding radiated waves are studied as functions of material properties. Four ferromagnetic	136 137 246	split Lf into two parts (due to dynamic and static magnetic fields). They numerically compared the 2.1. Step 1: Computing 𝑱 𝑒 , 𝑴 and 𝝐 𝑚𝑠
	194	materials (i.e., nickel and three grades of steel) are treated.	138 247	contributions of these mechanisms as functions of current The first step deals with the calculation of induced eddy
	195	These have similar elastic but contrasting electromagnetic	139 248	for ten steel grades. The dynamic Lf was found to be the currents (𝑱 𝑒 ), magnetization (𝑴) and magnetostriction
	196 197	properties. In Sec. 4.2 WGMs, are studied as functions of EMAT excitation (i.e., the current in the coil, and the	140 141 249 250	dominant mechanism at high currents, while the magnetostriction contribution depended strongly on the strain (𝝐 𝑚𝑠 ) from radiated fields (𝑬, 𝑯 𝑑 and 𝑯 𝑠 ). We start with the simpler case of conductive nonmagnetic materials
	198	static magnetic field of the magnet). Sec.5 summarizes	142 251	material. Although EMATs were used to generate various in which only eddy currents are induced. To compute
	199	the main findings of the paper.	143 252	types of waves, only a few works compared the various them, the solver CIVA-ET [28] is used. It operates under
	2. 201	144 145 146 253 254 255	WGMs to each other and linked them to the type of generated elastic wave(s). Moreover, when comparison took place, magnetic forces (volume and surface) were ISH and uses analytic solutions for 𝑯 𝑠 , while the finite element method is used for computing 𝑬 and 𝑯 𝑑 . Both the coil and a region of interest in the test piece are
	202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233	An EMAT is an EM source made of a coil (or of several coils) and a magnet (or several magnets) that radiates dynamic fields (electric (𝑬) and magnetic (𝑯 𝑑 )) and a static magnetic field (𝑯 𝑠 ). The dynamic source is an alternative current (𝑰 𝑒 ) circulating in the coil(s), and the static source is the magnet(s). 𝑬 induces eddy currents (𝑱 𝑒 ) in electrically conductive materials. These currents give rise to Lorentz forces in the presence of magnetic fields. 𝑯 𝑑 and 𝑯 𝑠 induce a magnetization (𝑴) in magnetic materials. As a consequence, magnetic forces are exerted on the material by an EMAT and a magnetostriction strain (𝝐 𝑚𝑠 ) is also induced. Owing to the dynamic nature of 𝑬 and 𝑯 𝑑 , these forces and strain create local dynamic strain/displacement in the part of the medium near the EMAT, thus radiating elastic waves in the medium. Throughout the present work, the infinitesimal strain hypothesis (ISH) is maintained: the strain associated with the elastic wave is sufficiently small for EM fields to be accurately computed on the undeformed shape of the part. When valid, ISH allows for the decoupling of electromagnetic problems (Steps 1 and 2 in what follows) and the elastodynamic problem (i.e., the wave radiation problem: steps 3 and 4). ISH is almost always accepted in the case of EMAT on conductive nonmagnetic materials since the wave displacement (few 𝑛𝑚 [32]) is at least three orders of magnitude smaller than the characteristic length of the elastic wave source (given generally by the skin depth (𝜇𝑚)). However, the case of ferromagnetic materials is different as it involves a static initial deformation due to magnetic force and magnetostriction strain (caused by the presence of the magnet). As a result, ISH is checked in the present work in the case of FM	147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 274 275 276 277 278 279 280 281 282 283 284 285 286	deemed insignificant to study or overlooked by the authors. While this does not pose a problem in specific configurations -as seen in Thompson [23] -it should not be regarded as the general rule. The present work aims to propose a unified model for all three mechanisms of electromagnetic-acoustic transduction in ferromagnetic media and to use this model to highlight, through parametric studies, the need to consider all three. Our findings are presented as follows. In Sec.2, the theoretical model of elastic wave generation (WG) by EMAT in FM is derived. The WG problem is solved in four steps. In the first (Sec.2.1), the electromagnetic problem consisting in obtaining eddy currents, magnetization and magnetostriction from radiated EM fields is solved under the assumption of infinitesimal strain hypothesis (ISH). Such a solution incorporates magnetic and magnetostrictive constitutive laws that were either measured or obtained following multi-scale (Daniel et al. [26]) or phenomenological (Jiles [27]) approaches into the solver CIVA-ET [28]. In Sec.2.2, elastic WGMs (volume force distributions) are given in terms of quantities obtained in Sec.2.1. Here, we follow the works of Bossavit [17] and Henrotte et al [29] to present Maxwell stress tensor method as an adequate tool to formulate EMAT wave generation mechanisms in ferromagnetic materials. Such formulation is carried out under the piezomagnetic behavior hypothesis (PMH). In Sec.2.3, the volume WGMs are converted -for computational purposes -into equivalent surface stress distributions using a previously developed tool by Clausse and Lhémery [30]. In the last step (Sec.2.4), the elastodynamic problem of calculating the elastic wave meshed. It is predictive as it uses only parameters obtained from stress-free measurements to predict behaviors under stress. Conversely, that proposed by Jiles [27] is a macroscopic phenomenological model, requiring parameters obtained from measurements under stress. To prove the premise of the present work (i.e., all WGMs should be included when designing an EMAT), materials need not be under stress, and hysteresis can be neglected. As a result, only stress-free versions of magneto-mechanical models are used. Within this framework, that by Daniel et.al. [26] uses the following three material parameters to describe the anhysteretic behavior: the initial magnetic susceptibility (𝜒
	234	materials.	179	radiation from the surface stress distributions is solved
			180	semi-analytically (based on Lhémery [31]), using the

This region is defined as the volume in which 𝑬 256 and 𝑯 𝑑 are considered nonnegligible. It represents the 257 location of the elastic wave source previously defined. The 258 case of FM materials is more complex and requires the 259 definition of magnetic and magnetostrictive constitutive 260 laws (𝑴(𝑯) and 𝝐 𝑚𝑠 (𝑯)). These laws can be obtained 261 experimentally or theoretically. In the present work, when 262 not obtained experimentally, they are either assumed 263 linear or obtained using one of the two following models: 264 the simplified multi-scale model (SMSM) by Daniel et al. 265 [26] (derived from the full model [34]) and the 266 phenomenological model by Jiles [27]. In their full forms, 267 they model hysteretic and anhysteretic magnetic and 268 magnetostrictive constitutive laws as functions of both 269 applied magnetic field and mechanical stress. Daniel et al. 270 model [34] is derived from first principles and treats the 271 magneto-mechanical problem in a bottom-up fashion, 272 starting at the domain scale and working its way-through 273 homogenization schemes -up to the polycrystal scale. 0 ), and both the 287 magnetization (𝑀 𝑠 ) and the magnetostriction strain (𝜆 𝑠 ) 288 at saturation. That by Jiles. [27] uses five material 289 parameters

  Step 2: computing 𝒇 𝑒𝑚 , 𝒇𝑚𝑠 and 𝒕 𝑚 304 Once the quantities (𝑱 e , 𝑴 and 𝝐 𝑚𝑠 ) are computed, they 305 are converted into body forces. The forces in question are 306 the electromagnetic force 𝒇 𝑒𝑚 (combination of Lorentz 307

	299	. Expressions of magnetic and
	300	magnetostrictive constitutive laws are then incorporated
	301	into the solver CIVA-ET [28] to obtain the induced eddy
	302	currents (𝑱 𝑒 ), magnetization (𝑴) and magnetostriction
	303	strain (𝝐 𝑚𝑠 ).
		2.2.

  𝑢 𝑘 (𝒙) = ∫ 𝑮 𝑘 (𝒙, 𝒙 𝟎 ) ⋅ 𝒇(𝒙 𝟎 )𝑑Ω 𝛿 𝒙 𝟎 ∈Ω 𝛿 Applying this conversion procedure to both 𝒇 𝑒𝑚 and 𝒇 𝑚𝑠 388 yields two equivalent surface distributions 𝝈 𝑒𝑚 and 389 𝝈 𝑚𝑠 .The total surface distribution is finally given as the 390 sum of three terms of surface stress as 𝝈 𝑡𝑜𝑡 = 𝝈 𝑒𝑚 + 391 𝝈 𝑚𝑠 + 𝒕 𝑚 . 392 2.4. Step 4: computing radiated elastic wave(s) 𝒖 393

				(9)
		∀𝒙 ∈ Ω, 𝑢 𝑘 (𝒙) ≈ ∫	𝑮 𝑘 (𝒙, 𝑿 𝟎 ) ⋅ 𝝈 ̃𝒇(𝑿 𝟎 )𝑑Γ	(10)
			𝑿 𝟎 ∈∂Ω 𝛿
	379	Where	Ω, Ω 𝛿 and 𝜕Ω 𝛿 denote respectively, the
	380	displacement field, the test piece volume, the volume in
	381	which the force distribution (𝒇) is nonnegligible and its
	382	surface. The depth of Ω 𝛿 is given by a few skin depths.
	383	𝑢 𝑘 (resp. 𝑮 𝑘 ) is the k-th component of the displacement
	384	field (resp. row of Green's tensor). Expressions of the
	385	equivalent surface stress distributions 𝝈 ̃𝒇 as functions of
	386	moments of 𝒇 and elastic constants are cumbersome and
	387	are provided in [30,33].

Table 1 .

 1 EM properties for the two test pieces.

	Test piece	Steel	Aluminum
	Electric conductivity (MS/m)	11.2	35
	Relative magnetic permeability	100	1

483 Maximum values correspond to EMAT coil center, while 484 null values correspond to its edges, where the magnetic 485 field is predominantly horizontal. Beyond the edges, the 486 normal component of the magnetic field is again 487 predominant. A little further from the coil, EM fields 488 vanish. The very good agreement between measured and 489 simulated signals (assuming ISH) proves the ISH to be 490 valid in this configuration. As a result, both problems -of 491 electromagnetic nature -treated in step 1 and 2 can be 492 dealt with separately from the problems -of 493 elastodynamic nature -treated in step 3 and 4. Such 494 separation allows the use of two different specialized 495 numerical solvers (i.e., CIVA-ET and CIVA-UT). 496

Table 2 .

 2 Material properties for the four FM materials.

	Material	𝜌 (𝑔/𝑐𝑚 3 )	𝐶 𝐿 (𝑚/𝑠)	𝐶 𝑇 (𝑚/𝑠)	𝜎 𝑒 (𝑀𝑆/𝑚)	𝜇 𝑖
	LCS	7.8	5900	3230	1.39	2
	Z20C13	7.89	5790	3100	2	15
	AISI410	7.67	5590	3000	2	10
	Nickel	8.88	5630	2960	15.4	110

  7 822 (Sec.3.3), since magnetic behavior is rarely linear. 823 Complementary studies were carried out considering 824 nonlinear behavior. Their conclusion regarding the premise 825 of the present work remains the same as that for linear The proposed method to solve WGP offers a framework 853 to predict WGMs as functions of material properties, 854 EMAT geometry, and excitation. Such predictions are 855 required in EMAT design for optimal elastic wave 856 radiation in ferromagnetic media. The method is readily 857 usable to deal with pre-stressed ferromagnetic materials, 858 which is the subject of future work. 859

	826
	827	behavior.
	828	5. Conclusion
	829	A solution to the problem of elastic wave generation
	830	(WGP) by EMAT in ferromagnetic materials has been
	831	proposed. The tools used include magnetic and
	832	magnetostrictive constitutive laws and numerical solvers
	833	for both electromagnetic induction and elastic wave
	834	radiation problems. The overall method is based on the
	835	two fundamental hypotheses of infinitesimal strain and
	836	piezomagnetic behavior. Both hypotheses were verified by
	837	experiments. The overall model was used to assess the
	838	validity of the premise of this paper, that all WGMs
	839	(electromagnetic force, magnetostriction strain, and
	840	magnetic traction) should be considered when designing
	841	an EMAT for applications involving ferromagnetic media,
	842	as none is always negligible. To this end, parametric
	843	studies were carried out to evaluate WGMs as functions
	844	of material properties and EMAT excitation. For a given
	845	EMAT, various excitations and four ferromagnetic
	846	materials with contrasting electromagnetic properties
	847	(nickel, AISI410, Z20C13, and low-carbon steel) were
	848	considered. These studies lead to the conclusion that
	849	WGMs depend greatly on material properties and EMAT
	850	excitation. A combination of material properties and
	851	excitation yields a predominant WGM that is negligible for
	852	another combination.

J. Acoust. Soc. Am. 96 (1994) 3787-3791.

Field maps (of maximum amplitude of a given quantity) are presented in Fig. 11. Due to lack of space, and the intimate correlation between maps and previously studied WGMs spatial distributions, only results of AISI410 are shown. Each WGM and the total distribution were treated separately. Whilst S-wave maps show all WGMs to be dominated by EM force, those for L-wave have contributions of comparable magnitudes. Curiously EMforce treated separately leads to a displacement amplitude (for S) larger than that of 𝝈 𝑡𝑜𝑡 ; this illustrates the fact 768 that WGMs do not always interact constructively. The 769 present result further confirms the assertion that the rule 770 when designing an EMAT or trying to understand EMAT 771 measurements is to evaluate each generation mechanism 772 contribution and compare their respective vectorial 773 components, as each component can give rise to a 774 different wave type. 775

Influence of EMAT excitation

In this subsection, we are interested in WGMs as functions of EMAT excitation (𝐼 𝑒 and 𝐵 𝑠𝑧 ). Given that in the present EMAT configuration, magnetic and magnetostrictive properties manifest themselves mostly through the normal component of WGMs, only this component is considered, for 𝐼 𝑒 ∈ [0; 100]A and 𝐵 𝑠𝑧 ∈ [0; 4]T, while the frequency is kept constant and equal to 1 MHz. The normalized maximum amplitude of each WGM is plotted as a function of 𝐼 𝑒 for LCS in Fig. 12a, and as function of 𝐵 𝑠𝑧 for LCS in Fig. 12b, for Ni in Fig. 12c, and for Z20C13 in Fig. 12d. The linear dependency of these maxima on current intensity is expected since the model was developed under the hypothesis of piezomagnetic behavior. Such an assumption is valid even at high intensities, as shown by Fig. 2, (for 𝐼 𝑒 = 175 A). This is also the case for the rest of the materials, however, due to the lack of space, only one material is presented here. The normal dynamic magnetic induction is around 100 mT, which is fifteen times smaller than that of the static one. This linear dependency was also observed for the other materials (whose results are not shown). For the material with the weakest magnetic and magnetostrictive properties (LCS), the electromagnetic force -specifically Lorentz force -dominates. Looking now at Fig. 12b, one notices that 801 magnetostriction WGM decreases non-linearly with 802 increasing 𝐵 𝑠𝑧 . This is due to the nonlinear 803 magnetostrictive behavior (Fig. 8), which shows that for 804 high 𝐵 𝑠𝑧 , magnetostriction saturates, leading to 𝒅 𝑚𝑠 = 𝟎 805 (see Eq.4). The divergence at low 𝐵 𝑠𝑧 is due to PMH being 806 invalid in this region. For this material (weak magnetic 807 properties), electromagnetic WGM -specifically Lorentz 808 force -dominates that of magnetic traction. The cases of 809 nickel (Fig. 12c) and Z20C13 (Fig. 12d) have similar 810 tendencies. The former shows, once again, that high 811 magnetic permeability and electric conductivity do not 812 necessarily lead to strong electromagnetic force, because 813 of the relatively small skin depth. The latter shows that 814

WGMs do not always interact constructively, as evidenced 815 by the dip of 𝜎 𝑧 𝑡𝑜𝑡 (bellow 𝜎 𝑧 𝑚𝑠 ) at around 𝐵 𝑠𝑧 = 0.7 T. 816