NDT&E International 51 (2012) Contents lists available at SciVerse ScienceDirect. NDT&E International

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1 NDT&E International 5 (202) 8 5 Contents lists available at SciVerse ScienceDirect NDT&E International journal homepage: The impact of magnetostriction on the transduction of normal bias field EMATs R. Ribichini a,n, P.B. Nagy a,b, H. Ogi c a UK Research Centre in NDE, Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK b School of Aerospace Systems, University of Cincinnati, Cincinnati, OH 4522, USA c Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan article info Article history: Received 7 March 202 Accepted 8 June 202 Available online 27 June 202 Keywords: Electromagnetic Acoustic Transducers Magnetostrction Lorentz force Steel abstract A very typical and important application of Electromagnetic Acoustic Transducers (EMATs) is the inspection of ferritic steels with normal bias field transducers. In this case, a controversy has arisen in the literature, as some older studies have indicated the Lorentz force as the main transduction mechanism, while more recent research has claimed that magnetostriction can be two order or magnitudes larger than the Lorentz effect. This is not merely an academic issue, as depending on which physical phenomena dominates, the performance of EMATs on different steel grades might significantly vary and the design of the transducer could be optimized accordingly. This paper analyzes in depth two main assumptions made in the more recent studies, highlighting some inconsistencies. A previously experimentally validated Finite Element model, is used to test the controversial assumptions. It is demonstrated that the mechanical boundary conditions were not modelled correctly leading to a gross overestimation of the role of magnetostriction. The main conclusion is that the magnetostriction force is typically not order of magnitudes larger than the Lorentz force; actually the Lorentz force is the larger transduction effect in non-oxidized ferromagnetic steels, and magnetostriction is only a fraction of it. & 202 Elsevier Ltd. All rights reserved.. Introduction Electromagnetic Acoustic Transducers (EMATs) have drawn significant attention in the Non-Destructive Testing community due to their advantages over traditional piezoelectric transducers. The contactless nature of EMATs overcomes one of the main limitations of standard piezoelectric probes: the need for a couplant liquid between the sample and the transducer. This is not only practically inconvenient in field conditions, but it also makes performing reproducible measurements extremely difficult. This can be a serious limitation especially when the amplitude of the signal has to be consistent over long periods of time, e.g. in Structural Health Monitoring applications, or over a spatial region, for example in arrays made of several elements. Another advantage of EMATs is the fact that their geometrical configuration can be designed in several ways, allowing to generate a wide range of wave-modes, including shear horizontal (SH) waves in plate-like structures or torsional waves in pipe-like components [,2]. The main disadvantages of EMATs are the relatively low signal-to-noise ratio compared to standard transducers and the n Corresponding author. Tel.: þ address: remo.ribichini06@imperial.ac.uk (R. Ribichini). fact that their performance can depend significantly on the material properties of the inspected sample. In order to overcome these issues and design better transducers, several researchers [3 5] have analyzed the transduction mechanisms that allow EMATs to generate and detect ultrasonic waves. The most complex case is the operation of EMATs on ferromagnetic samples, like ferritic steel, where three distinct phenomena occur and contribute to the transduction: the Lorentz force, which occurs in any electrically conducting material, and magnetostriction and the magnetization force, that take place only in ferromagnetic media. The different transduction phenomena are influenced by material properties (like electric conductivity or magnetic permeability) and operational conditions, such as the driving frequency or the magnetic bias field, in very different ways. For instance, the Lorentz force mechanism is linear and not very sensitive to electromagnetic properties (at least in good electrical conductors). In contrast, magnetostriction is highly non-linear and its contribution to wave generation and reception depends significantly on the physical properties of the sample and on the magnetic bias field. For these reasons, it is paramount to assess which transduction mechanism dominates the operation of a given EMAT configuration since this affects the performance of the transducer when used on different materials. Previous studies /$ - see front matter & 202 Elsevier Ltd. All rights reserved.

2 R. Ribichini et al. / NDT&E International 5 (202) have convincingly established that magnetostriction is the main transduction mechanism for EMATs whose magnetic bias field is parallel to the surface of the sample [5]. However, several other EMAT configurations employ a static bias field normal to the sample, for example to generate shear bulk waves or Lamb waves in plate-like structures [6,7]. In these cases, there seem to be a controversy in the literature, since some authors claimed that the Lorentz force is the dominant mechanism [5,8,9], while a more recent study by Ogi [0] came to the conclusion that magnetostriction is the more significant effect even in the case of nonoxidized steels. Based on the erroneous analysis of [0], a widely used monograph on the scientific and industrial applications of EMATs, that was subsequently published by Hirao and Ogi, claimed that magnetostriction can be as much as two orders of magnitude larger than the Lorentz force in ferritic steels []. This is not only an academic question, because according to which physical phenomenon dominates, the design of the transducer could be optimized accordingly and very different performance could be observed in field applications. This paper corrects the erroneous approximations introduced in [0] by one of the authors of the present paper. It will be shown that for the typical case of a bulk shear wave EMAT with normal bias field some of the assumptions made in those studies are not applicable and lead to a significant overestimation of the effects of magnetostriction. Two main assumptions are analyzed in detail and compared with novel analytical relationships as well as with a previously validated Finite Element model [2]. The results of this comparison are used to resolve the existing discrepancy in the literature, and conclusions are drawn. 2. EMAT theory In order to assess the assumptions made in [0] first we have to present the general principles that govern EMAT operation. These are the basic equations that underly EMAT physics; the discrepancy in the literature arises from the applications of these principles to specific cases, i.e. from the application of the correct boundary conditions and the use of realistic physical parameters. In any electrically conducting material the Lorentz force per unit volume f L is produced by the interaction of an eddy current density, J e, with a static magnetic flux density, B: f L ¼ J e B: The static magnetic field can be produced by a permanent magnet or electromagnet; the eddy currents are induced by a driving current fed to a coil during the wave generation process, while during the reception phase they are induced by the movement of a conducting solid through a stationary magnetic field. Eq. () implies that the Lorentz force is linear in J e and B. Furthermore, it has been shown [5] that in the high-conductivity limit, i.e. when the electromagnetic penetration depth is much smaller than the acoustic wavelength, the Lorentz force is relatively insensitive to material properties such as the electric conductivity s and magnetic permeability m. This is a consequence of the shielding effect that occurs in highly conductive materials: the eddy currents tend to equal and mirror the driving current, regardless of their spatial distribution which is governed by conductivity and permeability [5,3]. The other main transduction mechanism, magnetostriction, occurs only in ferromagnetic media and is similar to the piezoelectric effect but, the mechanical strain is induced by a magnetic field rather than by an electric field [3 5]. Physically this is due to the fact that magnetic spins and domains (depending on the field intensity) tend to align parallel to the total magnetic field, resulting in a net mechanical deformation with no change in ðþ volume (at least below the Curie temperature). Differently from the Lorentz force, magnetostriction is non-linear: the characteristic curve strain, E t, versus magnetic field, H (also known as magnetostriction curve), depends on the given material, shows some degree of hysteresis and depends on the stress state, the magneto-mechanical loading history and surface conditions [2,4]. In practice, in EMAT modelling, magnetostriction can be often linearized. This is due to the fact that EMATs normally employ strong permanent magnets or electromagnets that generate a static magnetic field H which is much larger than the dynamic magnetic field H, ~ caused by the current flowing in the coil. Throughout this paper we designate static quantities with a bar, as opposed to dynamic ones, designated with a tilde. When H b H, ~ there are only small oscillations of the total magnetic field (i.e. the vector sum of H and H) ~ around the operation point H, which can be approximated with a linear relationship [6]: ( ~E ¼ S H ~s þd H, ~ ~B ¼ d T ~s þl s H, ~ ð2þ where E and s are the strain and stress tensors (in abbreviated vector form) and B and H are the magnetic flux density and the magnetic field strength respectively; S H is the elastic compliance matrix measured with constant H; l r is the magnetic permeability matrix at constant stress. The magnetostriction matrix d (6 3) describes the interaction between the magnetic field and the elastic field. It has been shown [7] that within the strong bias approximation d has only two independent components that can be derived from the magnetostriction curve. The first independent component, d 22, is related to the changing magnitude of the magnetostrictive strain and therefore proportional to the derivative of the magnetostriction curve at the operation point, while the second parameter, d 6, is related to the changing direction of the magnetostrictive strain and therefore proportional to the ratio of the magnetostrictive strain, E t, and the magnetic bias field: d 6 ¼ 3E t H : It can be shown [] that this parameter is the most important in bulk shear wave generation when the bias field is normal to the surface of the sample. Both the Lorentz force and magnetostriction have mechanical to electromagnetic counterparts, i.e. ultrasonic waves in the testpiece induce an electrical signal in the transducer so it can be used as a receiver. Finally, it should be mentioned that a third transduction mechanism exits on ferromagnetic materials: the magnetization force. This mechanism can generate oblique incidence bulk shear waves [9], but it is widely agreed that its contribution to bulk waves propagation normally into the sample is negligible and will not be included in the subsequent analysis. In his original paper [0], Ogi used Maxwell s equations to derive the dynamic magnetic field induced by an elongated spiral coil in a ferritic steel sample. The problem was modelled in a twodimensional space, half of which was occupied by air, while the other half-space was filled with a ferromagnetic medium. The coil was modelled as a tangential current sheet driven by a timeharmonic excitation. Once both the dynamic and static magnetic fields were computed, Ogi estimated their mechanical effects, by applying Eq. () for the Lorentz force and deducing an equivalent force for the magnetostriction effect. As we shall see in the following sections of this paper, in the process of converting the effect of the magnetostriction strain into a body force, Ogi erroneously modelled the mechanical boundary conditions, thus causing an overestimation of magnetostrictive wave generation. A similar misjudgment was also done in the electromagnetic ð3þ

3 0 R. Ribichini et al. / NDT&E International 5 (202) 8 5 boundary conditions. The analysis of these approximations is presented in the following section. 3. Boundary effects In order to compare the new analysis of magnetostriction generated waves presented in this paper with the previous one given in [0] we study the same EMAT configuration and use the same notation as the original reference. We refer to a twodimensional model in the x x 3 plane, where the semi-space x 3 40 is filled by a ferromagnetic material, while the other semispace is made of air (Fig. ). A permanent magnet or electromagnet provides a static bias field H 3 normal to the surface. The EMAT coil is modelled as a surface current sheet, parallel to the sample and infinitely wide ðb-þ and at a distance h (lift-off) from the ferromagnetic medium. The coil of n turns per unit length is fed by an oscillating current Ie iot, where i is the imaginary unit, o is the driving angular frequency and t is time. The driving current induces eddy currents in the sample, decaying with depth in an exponential fashion. The tangential component of the magnetic field, H ~ within the material ðx 3 40Þ can be described as [3] ~H ¼ H ~ 0 e ikx 3, ð4þ where H ~ 0 is the p maximum amplitude of the dynamic field at the surface and k ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ioms ¼ð iþ=d, with d standard penetration depth. In order to generate bulk shear waves, forces tangential to the surface must be generated. Using Eqs. () and (4) it can be found that the Lorentz force mechanism generates shear waves mainly due to a body force [6]: f L ¼ H ~ 3 ¼ ikb 3 H ~ : 3 On the other hand, by combining Eqs. (2) and (3) it can be deduced that, within the strong bias field approximation, the magnetostrictive shear strain is ~E 3 ðx 3 Þ¼d 6 ~ H ðx 3 Þ¼ 3E t H 3 ~ H ðx 3 Þ: 3.. Mechanical BCs In the original Ref. [0], the result obtained in Eq. (6) is directly combined with the standard elastic constitutive equation for an infinite isotropic medium: s 3 ¼ c 55 E 3 to get the shear stress, µ 0 h ~ H (x 3 = 0 - ) µ 0 µ r ~ H (x 3 = 0 + ) b x 3 J = ni J e (x 3 ) Air Sample Fig.. EMAT model investigated in the paper. The coil is modelled as a surface current sheet, parallel to the sample and infinitely wide ðb-þ and at a distance h (lift-off) from the ferromagnetic sample. A magnet (not shown) provides a static bias field H 3 normal to the surface. Eddy current, J e, exponentially decaying with depth is induced in the sample. x ð6þ where c 55 is the elastic shear stiffness. Then, this stress is used in the equilibrium equation: = rþf ¼ r u, ð7þ where r is the mass density, u is the displacement and f is the body force. In the original reference the inertial term r u was neglected and the main body force component causing shear waves was simply replaced with: f original 3 : 3 Substituting the expression for s 3 into Eq. (8), it was found that f original ¼ 3E tc H ~ : H 3 Using Eq. (4) we finally find f original H ~ ¼ 3ikE t c 55 : ð0þ H 3 Substituting into this equation the typical parameters for bulk wave EMAT inspection of a low carbon steel, e.g. H 3 ¼ 5kA=m, c 55 ¼ 80 GPa and E t ¼ 2 ppm, the authors erroneously concluded that the tangential force due to magnetostriction is a hundred times larger than the Lorentz force [0,]. Magnetostrictive strain is what generally called in the literature eigenstrain [8]. The best known example of eigenstrains is thermal expansion. Uniform eigenstrains do not generate any stress in an elastic body unless the material is constrained somehow. If a homogeneous, isotropic material is absolutely free of external constrains, eigenstrains do not generate stresses at all. Equivalent body forces should be based on a true body force, like inertia force. Alternatively, if the original bulk body force of Eq. (0) is used, like it was by the author in [0], an artificial surface traction must be introduced in the wave amplitude calculations to assure that the surface remains traction free. The traction-free boundary condition was considered but not properly satisfied in [0]. The error was caused by treating the magnetostrictive bulk body force of Eq. (0) as if it were a true body force like the Lorentz force, i.e. independent of surface tractions, which it is not. Actually, the r u inertia force is the only body force that leads to bulk wave generation in the material. This leads to rewriting the tangential component of the body force, f as f ¼ 2 ¼ ro2 u : ðþ Physically, only a small portion of the ferromagnetic medium near the surface is being deformed, with a shear strain given by Eq. (6). The displacement due to this deformation is approximatively given by spatial integration of ~E 3 ðx 3 Þ: Z H0 ~ e u ðx 3 Þ¼ ~E 3 dx 3 ¼ 3E ikx3 t x 3 H 3 ik : ð2þ Substituting Eq. (2) into (), the inertial force caused by magnetostriction is found: f ¼ 3E to 2 r ~H : ð3þ ik H 3 We can finally assess the ratio between the original approximated tangential magnetostriction force (Eq. (0)) and the magnetostriction force obtained with the new analysis (Eq. (3)): f f original ¼ or ¼ k s 2 msc 55 k, ð4þ p where k s ¼ o=c s is the shear wavenumber and c s ¼ ffiffiffiffiffiffiffiffiffiffiffiffi c 55 =r is the shear wave velocity. Eq. (4) shows that the two analyses differ very significantly. The ratio of the predicted tangential body ð9þ

4 R. Ribichini et al. / NDT&E International 5 (202) 8 5 forces is proportional to the excitation frequency and mass density and inversely proportional to the magnetic permeability, electric conductivity and shear stiffness. With the Finite Element model that will be presented in the following section, Eq. (4) allows us to test which analysis is the correct one Electromagnetic BCs Another important approximation made in Ref. [] deals with the electromagnetic boundary condition between air and the ferromagnetic half-space. According to Eq. (.6) of [], the tangential component of the dynamic magnetic field, H ~ is continuous at the interface, as prescribed by Maxwell s equations, however, the dynamic magnetic field at the interface is assumed to be the same that would be produced in a homogeneous medium, i.e.: ~H original ðx 3 ¼ 0Þ¼ ni 2 ¼ J 2, ð5þ where I is the driving current, n is the number of coil turns per unit length and J is the current density per unit length. Actually, Eq. (5) is only valid within a homogeneous medium; the presence of an air sample interface, causes the induction of eddy currents and subsequently the magnetic field at the interface is very different from the one given in Eq. (5). In order to account for the electromagnetic interface several techniques can be used. A simple approach is the method of the magnetic images [9 2], where the magnetic field in air is computed as the superposition of the original source I with a reflection image I 0 placed in infinite air, and the magnetic field within the ferromagnetic material is computed as that caused by a transmission image I 00 placed in infinite ferromagnetic medium. The total dynamic magnetic field results from the superposition of the field due to the current sheet J¼nI flowing in the coil and the field generated by the eddy current, J e. The eddy current within the metal decays exponentially with depth, however, here we can substitute it with a current sheet, flowing in the opposite direction of the driving current (due to Lentz law) and placed at an arbitrary small depth below the surface that is less than the skin depth d. Since the interior of the metal must be shielded from the external excitation, the eddy current within the material must vanish for sufficient depth, therefore it can be imposed that J ¼ J e.withthismodel,we find that the magnetic field at the interface between air and a magnetic conductor is ~H ðx 3 ¼ 0Þ¼J: ð6þ parameters, with a 720% accuracy in the frequency range khz and over a wide range of bias field strength (0 80 ka/m). Moreover, the FE model directly employs the constitutive equations of magnetostriction (Eq. (2)), rather then equivalent forces, so it is independent from the approximations made in the previous section and can be used as a reliable reference. A two-dimensional, plain strain, model in a reference coordinate system fx,x 3 g of a normal bias field EMAT has been developed. The driving current in the coil is modelled as a zero cross-section current sheet, flowing in the out of plane direction (x 2 ). The coil lies 0.6 mm above the metal and induces eddy current J e in the x 2 out of plane direction. Magnetostriction contributes to the wave generation producing shear strains ~E 3 described by Eq. (6). Other magnetostriction contributions are also present but being proportional to the normal component of the dynamic field, H3 ~, they are considerably lower since the dynamic magnetic field H ~ below the coil is mostly parallel to the surface of the sample. The coil is 8 mm wide and is driven by a A current. The mesh consists of approximatively 00,000 triangular elements. The material properties of steel were used as a reference: Young s modulus 200 GPa, Poisson s ratio 0.33, mass density 7850 kg=m 3, electric conductivity 4 MS/m, normal relative permeability m r ¼ 60. In order to reduce the computational burden of the model, full magnetostrictive constitutive equations are employed only just below the coil where the induced eddy currents are concentrated. For a depth larger than a few skin depths d, i.e. 9x 3 944d, the dynamic magnetic field H ~ becomes negligible (Eq. (4)) and no transduction occurs. For this reason, purely elastic constitutive equation can be used to describe wave propagation saving significant computational time. In order to simulate the operation on a half-space, an absorbing region employing Rayleigh damping surrounds the elastic domain, to avoid back-reflections from the edges of the model. The result of a typical FE simulation is shown in Fig Permeability modelling A further issue in the magnetostriction model of Ref. [0] does not involve the analytical derivation of the model, but rather the material properties given as input. In particular, the values of the relative magnetic permeabilities of steel seem significantly lower than those found in the literature (Fig. 3). Using the magnetization curves of several sources [6,23 25] the magnetic permeabilities normal and parallel to the bias field were estimated. Within the Eq. (6) shows that the presence of an electromagnetic interface results in a dynamic field two times larger than the magnetic field that would be predicted for a homogeneous medium, as in Eq. (5). The error caused by the use of Eq. (5) will be assessed in Section 5 by means of Finite Element simulations. Magnet 0.5 [pm] 4. Computational method 4.. Finite element model An EMAT Finite Element (FE) model has been developed using a commercial package, COMSOL Multiphysics [2]. The software solves atthesametimetheelectrodynamicproblem,describingtheeddycurrent induction, and the mechanical problem, accounting for wave generation. The magnetostriction constitutive equations couple the electromagnetic field with the elastic field. The linearized magnetostrictive equations are employed under the assumption of a strong bias field. This magnetostriction model has been quantitatively validated for SH0 wave generation in nickel [2] and steel plates [22] for different EMAT configurations. The model managed to predict the wave amplitude from first principles, without any adjustable Coil x 0 x 3 λ =.5 mm Dynamic Magnetic Field Shear Wave -0.5 Fig. 2. FE model of the EMAT. The displacement in the x direction generated by the transducer is represented by the colour plot. The amplitude of the parallel component of the dynamic magnetic field, ~ H, produced by the coil is represented by the contour lines. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

5 2 R. Ribichini et al. / NDT&E International 5 (202) 8 5 Relative Magnetic Permeability, µ r 50 μ -Ogi, original μ - Ogi μ -[26] μ - Thompson μ -Bozorth 00 μ -Ogi, original μ - Ogi \ μ -[26] 50 0 large bias field approximation, when the static and dynamic magnetic fields are parallel to each other, only the magnitude of the total magnetic field, i.e. H ¼ H þ ~ H changes, while its direction is constant, thus the effective magnetic permeability is m J r m : ð7þ On the other hand, when the static bias field is perpendicular to the dynamic magnetic field, the total magnetic field tilts around an equilibrium point without changing its magnitude (in the first order approximation), thus the effective permeability can be computed as m? r ¼ B m 0 H : ð8þ In the considered EMAT configuration, the normal permeability plays a key role, as the static bias field and the dynamic field are approximatively perpendicular to each other near the surface of the sample, where most of the transduction occurs. Fig. 3 shows the permeability curves computed from magnetization curves taken from the literature [0,6,25,26] and using Eqs. (7) and (8) together with the values given in [0]. Although the effective dynamic permeability is arguably the most uncertain material parameter needed to model the transduction sensitivity of EMATs on ferromagnetic materials, the rather low values selected by the author in [0] do not represent the majority of ferritic steels. Since increasing permeability reduces the dynamic magnetic field produced by the EMAT coil in the material (and increases outside it), any underestimation in the permeability leads to an overestimation in the sensitivity. However, it is important to point out that both Lorentz and magnetostrictive transduction mechanisms are influenced in a rather similar fashion, therefore their relative contribution is only slightly affected. The effect of the underestimation of the magnetic permeability on EMAT wave amplitude will be shown in Section 5 through Finite Element simulations. 5. Results Static Magnetic Field, H [ka/m] Fig. 3. Permeability curves of low-carbon steels computed from magnetization curves from Ogi [0], Thompson[25], Bozorth[6], and[26]. Eqs. (7) and(8) were used to obtain the parallel (dashed lines) and normal permeabilities (continuous lines). The comparison of these values against those given in [0], shows that the latter are rather low and do not represent typical low-carbon steels. 5.. Mechanical BCs The Finite Element model described in the previous section has been employed to test the analytical approximations presented in Table Summary of the material properties used in the FE model. Conductivity (MS/m) Relative permeability Refs. [0,] and in this paper. The body force in the tangential direction, f, near the surface of the sample due to the shear bulk wave EMAT was computed with Eqs. (0) and (3). The obtained values were then compared with the FE predicted body force on the surface of the ferromagnetic medium just below the midpoint of the coil ðx ¼ x 3 ¼ 0Þ. According to Eq. (4), the discrepancy between the two analytical approximations is proportional to the mass density, r, and the driving frequency, o, and inversely proportional to conductivity, s, and permeability, m r. By sweeping through a wide range of these parameters, it can be clearly seen whether Eq. (0) or (3) is the correct one. Starting from the material properties of mild steel (central row of Table ), each of the mentioned parameters was in turn reduced to /2 and /4 of the original value or doubled and quadrupled while keeping all the other parameters constant. For instance, the range of electric conductivities f; 2,4; 8,6g MS=m was input while all the other parameters were kept constant and equal to the nominal ones. In this way a significant dynamic range (24 db) of four different parameters was investigated. A summary of the parameters used in this analysis is given in Table. The results of this study are given in Fig. 4. In each sub-figure, the forces computed with Eqs. (0) and (3) are divided by the corresponding FE result, taken as the reference, and plotted against the four investigated parameters ðr,o,s,m r Þ. The agreement between the new analysis, i.e. Eq. (3), and the Finite Element model is excellent: the difference between the predictions of the two methods is at most 6%. On the other hand, it is clear that the analysis of Refs. [0,] leads to large discrepancies with the FE model, in the order of two or three orders of magnitude over the investigated ranges. The disagreement of the old analysis with the FE results is proportional to r and o, and inversely proportional to s and m r, in line with Eq. (4). For the nominal physical properties of steel and driving frequency 0.5 MHz (central row of Table ), Eq. (0) erroneously predicts a tangential force ffi000 times larger than the force predicted with the new analytical approximation and by the Finite Element Method Electromagnetic BCs Driving frequency (MHz) Mass density (kg/m 3 ) , ,400 By using Eq. (5) to calculate the dynamic field at the surface of the sample, the magnetic field produced by the coil in a homogeneous medium is obtained. In comparison, Eq. (6) represents the tangential magnetic field that satisfies the boundary conditions between a dielectric medium and a good conductor. This formula takes into consideration the reflection that occurs at the interface that essentially doubles the magnetic field in the conductor. The presence of two different materials with different conductivity and permeability creates a boundary that affects the dynamic magnetic field. The dynamic magnetic field predicted with Eqs. (5) and (6) is shown in Fig. 5. These theoretical values refer to an infinitely wide coil and thus they do not depend on the magnetic permeability of the sample, i.e.

6 R. Ribichini et al. / NDT&E International 5 (202) Conductivity, σ [MS/m] Relative Magnetic Permeability, µ r Density, ρ [Kg/m3] Frequency, f [MHz] Fig. 4. Validation of the analytical models proposed in [0] (crosses) and in the present paper (triangles), against the Finite Element model. The magnetostriction forces computed with Eqs. (0) and (3) are divided by the corresponding FE result, taken as the reference, and plotted against the four investigated parameters: (a) electrical conductivity, s, (b) relative magnetic permeability, m r, (c) mass density, r and (d) driving frequency, f. A summary of the parameters used in this analysis is given in Table. Dynamic Magnetic Field, H [A/m] they are represented by horizontal straight line in the graph. The FE model was used to evaluate the discrepancy between Eqs. (5) and (6). The same geometry and properties described in Section 4. were used. When we choose to input the s air ¼ s sample and m air r 00 H = J FE with σ air =σ sample and µ r = 0 H = J / 2 b / h = 200 b / h = 40 b / h = 00 b / h = Relative Magnetic Permeability, µ r Fig. 5. Dynamic magnetic field, H ~, produced by an EMAT coil at the surface of the sample. The two horizontal lines H ¼ J=2 and H¼J represent the analytical predictions from Eqs. (5) and (6) respectively. These predictions assume an infinitely wide coil, i.e. ðb=hþ-. The curved lines are results from the numerical FE model for four finite size coils with different ratios width over lift-off distance, (b/h). The square represents an FE simulation for s air ¼ s sample and m air r ¼ m sample r ¼ (homogeneous medium with no interface). ¼ m sample r ¼ to the numerical model we obtain a tangential magnetic field represented by the square in Fig. 5. This matches with the predicted value of Eq. (5), since we are essentially describing the case of a coil surrounded by a homogeneous infinite medium. When an interface is introduced because of the presence of a conductor with s sample ¼ 4MS=m, and m sample r A½; 00Š, the magnetic field predicted by the numerical model is significantly different from that given by Eq. (5) and is a function of the magnetic permeability as well as the ratio between the coil width and the lift-off distance, (b/h). Fig. 5 shows four configurations from (b/h)¼20 to (b/h)¼200. The numerical predictions show that the larger the ratio (b/h), the more we approach the limit case ðb=hþ-, that corresponds to an infinitely wide coil, and thus matches with the analytical expression of Eq. (6). The exact analytical expression that takes into account the electromagnetic properties of the sample (permeability and conductivity), as well as the lift-off distance and size of the coil can be found, for example, in Chari and Reece [27]. Using such expressions an excellent agreement with the Finite Element simulations can be found; however, as far as we are concerned in this paper, i.e. the ideal case of an infinite coil, we can conclude that Eq. (6) correctly predicts the dynamic magnetic field, while the use of Eq. (5) leads to an error of a factor of 2. Since both Eqs. (0) and (3) show that the resulting tangential force is directly proportional to ~H, we can conclude that by neglecting the ferromagnetic interface, Refs. [0,] have introduced an underestimation of the magnetostriction force of about a factor of Permeability The permeability curves of Fig. 3 show that both the parallel and normal permeabilities were significantly underestimated in Ref. [0]. The impact of permeability on magnetostrictive wave generation has been analyzed in [2]: lower permeabilities cause a larger skin depth which means that there is a larger region where a significant dynamic field ~ H is present. Moreover, a low permeability implies that the mismatch between the permeabilities of air and the

7 4 R. Ribichini et al. / NDT&E International 5 (202) 8 5 Table 2 Permeabilities used for the FE simulations of Fig. 6: as given in Ref. [0] and with the new analysis, i.e. computed using Eqs. (7) and (8). Magnetic bias field, H 3 (ka/m) sample is smaller and thus the dynamic magnetic field is larger (Fig. 5). As a result, on two hypothetical materials with exactly the same properties, apart from permeability, a purely magnetostrictive EMAT would generate a higher signal amplitude in the material with lower m r. To assess this effect quantitatively, the Finite Element model described in Section 4. was run with different sets of permeabilities fm J r,m? r g: as given in [0], and computed using Eqs. (7) and (8). The permeabilities were estimated at three different levels of magnetic bias field H 3, in the range, H 3 Af5; 5g ka=m, which has been shown [28] to be a realistic range for normal bias field EMATs employing rare earth magnets. The permeabilities used are summarized in Table 2. As expected, the wave amplitudes obtained with lower permeabilities were larger than their counterparts with higher permeabilities, the difference between the two cases increases with the permeability mismatch. Fig. 6 shows that by using the magnetic permeability of Ref. [0] rather than those computed with Eqs. (7) and (8), the wave amplitude is overestimated by a factor between 3 and5intheconsideredbiasfieldrange. 6. Discussion Ref. [0] New analysis m? r m J r m? r m J r Displacement [arb.] Recomputed Permeability Permeabilty from Reference Static Magnetic Field, H [ka/m] Fig. 6. Finite element predicted displacement caused by magnetostriction using the same EMAT. Two different sets of permeabilities were used as an input to the model: as given in [0] (dark bars), and as computed using Eqs. (7) and (8) (grey bars). The permeabilities used were estimated at three different levels of magnetic bias field H 3, and are summarized in Table 2. The new analysis of normal bias field EMATs operating on ferromagnetic samples has highlighted two invalid assumptions made in previous research [0,] and the use of a low value of magnetic permeability. The impact of such assumptions on the predicted wave amplitude due to magnetostriction is very different. By neglecting the presence of an electromagnetic interface and underestimating the magnetic permeability, the resulting error is a reduction of the predicted wave amplitude by a factor of 2 in the former case and an overestimation by a factor between 3 and 5 in the latter case. These effects partly compensate each other, and overall they do not cause huge discrepancies with the FE predictions. On the other hand, if the mechanical boundary conditions are not modelled properly, as in Eq. (0), a very large error is introduced. The physical reason is that magnetostriction causes a deformation of a small portion of the material near the surface of the sample, whose depth is dictated by the electromagnetic skin depth, d. In a homogeneous elastic medium such a strain would induce large elastic forces, as wrongly predicted by Eq. (0). Actually, the presence of the boundary with air means that the surface of the sample must be stress-free, and the deformed layer is vibrating without transmitting large stresses into the bulk of the material. In other words, magnetostriction is only generating inertial forces proportional to the mass of the shallow subsurface layer of the material that it sets in motion Eq. (). We can further understand the situation by an analogy with laser generated ultrasound [29,30]. In the thermoelastic regime, the temperature of a thin subsurface layer is increased and generates thermal expansion. If the surface of the sample is traction-free, only inertial forces are generated resulting in weak pressure waves. On the contrary, if the surface is somehow constrained, for example by spraying water on it, the thermally expanded layer causes large elastic stresses and thus large pressure waves. In [0,], the stress-free boundary condition was not taken into account correctly, leading to the erroneous conclusion that for a normal bias field EMAT the magnetostriction effect is 00 times larger than the Lorentz force in a low-carbon steel. From the results of Fig. 4 we can conclude that magnetostriction was overestimated by a factor of , thus the actual ratio between the magnetostriction induced wave amplitude and that due to the Lorentz force is in the range A similar conclusion had been reached by experimental observation of the EMAT wave amplitude dependence on the electromagnetic properties of a range of different steel grades [28]. This result implies that, contrary to what was stated in [0,], magnetostriction is not the dominant transduction mechanism for normal bias field EMATs used on ferritic steel samples. On the contrary, the contribution of magnetostriction to wave generation is only a fraction of that due to the Lorentz force. From a practical point of view, since the Lorentz force is less sensitive to material properties than magnetostriction, it is possible to use a single normal bias field EMAT design to inspect different ferritic steel grades without experiencing large variations in the performance of the transducer. However, the results presented do not contradict the fact that magnetostriction is of the same order of magnitude or larger than the Lorentz force in highly magnetostrictive materials, such as nickel, iron cobalt alloys or ferrite oxides, or when the magnetic bias field is parallel to the surface of the sample [2,5,8,9]. 7. Conclusions When operating on ferromagnetic materials, electromagnetic acoustic transducers mainly exploit two transduction phenomena: the Lorentz force and magnetostriction. There is some controversy in the literature regarding which phenomenon dominates when the bias field is normal to the testpiece. In the case of non-oxidized ferritic steels, some researches stated that the Lorentz force is the main effect [5,8,9], while magnetostriction has a negligible contribution. More recent studies [0,] that have been accepted by many practitioners claim that magnetostriction can be as much as two orders of magnitude larger than the Lorentz force even in non-

8 R. Ribichini et al. / NDT&E International 5 (202) oxidized ferritic steels. This issue is of practical importance as depending on which transduction mechanism dominates, the performance of the transducer might dramatically change when inspecting different steel grades and the design of the transducer should be adjusted. In this paper, some of the assumptions made in Refs. [0,] were analyzed in depth from an analytical point of view. It was found that in those studies the influence of the traction-free surface was modelled in an inappropriate way, leading to a large overestimation of the effects of magnetostriction. It was also found that the electromagnetic boundary conditions were not taken into account correctly. These findings are supported by a previously experimentally validated Finite Element model. We also showed that the magnetic permeability was somewhat underestimated in [0,] that also influences the transduction sensitivity. It has been found that the Lorentz force is the strongest transduction mechanism on ferritic steels, in agreement with previous studies [5,8,9]. The wave amplitude due to magnetostriction is a fraction of that due to the Lorentz effect, and it has been definitely refuted that it is two orders of magnitude larger than the Lorentz force. As a consequence, since the Lorentz force is relatively insensitive to the range of material properties of steels, the same EMAT probe can be used on various steel grades with consistent performance. However, this does not exclude that when a highly magnetostrictive oxide or alloy is present on the surface of the sample magnetostriction becomes the dominant phenomenon. References [] Thompson RB. Generation of horizontally polarized shear waves in ferromagnetic materials using magnetostrictively coupled meander-coil electromagnetic transducers. Appl Phys Lett 979;34:75 7. [2] Kwun Hegeon, Teller Cecil M. Magnetostrictive generation and detection of longitudinal, torsional, and flexural waves in a steel rod. J Acoust Soc Am 994;96: [3] Dobbs ER. Electromagnetic generation of ultrasonic waves. In: Mason WP, Thurston RN, editors. Physical acoustics, vol. X. New York: Academic Press; 973. p [4] Maxfield BW, Fortunko CM. The design and use of electromagnetic acoustic wave transducers (EMATs). Mater Eval 983;4: [5] Thompson RB. Physical principles of measurements with EMAT transducers. In: Mason WP, Thurston RN, editors. Physical acoustics, vol. XIX. New York: Academic Press; 990. p [6] Kawashima K. Theory and numerical calculation of the acoustic field produced in metal by an electromagnetic ultrasonic transducer. J Acoust Soc Am 976;60: [7] Wilcox PD, Lowe MJ, Cawley P. The excitation and detection of lamb waves with planar coil electromagnetic acoustic transducers. IEEE Trans Ultrason Ferroelectr Freq Control 2005;52(2): [8] Wilbrand A. EMUS probes for bulk waves and rayleigh waves, model for sound field and efficiency calculations. In: Holler P, editor. New procedures in nondestructive testing. Berlin: Springer; 983. p [9] Wilbrand A. Quantitative modeling and experimental analysis of the physical properties of electomagnetic ultrasonic transducers. In: Chimenti D, Thompson D, editors. Review of progress in quantitative nondestructive evaluation, vol. 7. New York: Plenum; 987. p [0] Ogi H. Field dependence of coupling efficiency between electromagnetic field and ultrasonic bulk waves. J Appl Phys 997;82: [] Hirao M, Ogi H. EMATs for science and industry: noncontacting ultrasonic measurements. Boston: Kluwer Academic Publishers; [2] Ribichini R, Cegla F, Nagy PB, Cawley P. Quantitative modeling of the transduction of electromagnetic acoustic transducers operating on ferromagnetic media. IEEE Trans Ultrason Ferroelectr Freq Control 200;57(2): [3] Jiles D. Introduction to magnetism and magnetic materials. London: Chapman and Hall; 998. [4] Engdahl G. Handbook of giant magnetostrictive materials. San Diego: Academic Press; [5] Lee EW. Magnetostriction and magnetomechanical effects. Rep Prog Phys 955;8(): [6] Bozorth RM. Ferromagnetism. Princeton: Van Nostrand; 95. [7] Ogi H, Goda E, Hirao M. Increase of efficiency of magnetostriction sh-wave electromagnetic acoustic transducer by angled bias field: piezomagnetic theory and measurement. Jpn J Appl Phys 2003;42: [8] Qu J, Cherkaoui M. Fundamentals of micromechanics of solids. Hoboken: Wiley; [9] Hammond P, Sykluski JK. Engineering electromagnetism: physical processes and computation. Oxford: Oxford Science Publications; 994. [20] Weber E. Electromagnetic fields. New York: J. Wiley & Sons, Inc.; 972. [2] Demarest KR. Engineering electromagnetics. USA: Prentice-Hall; 998. [22] Ribichini R, Cegla F, Nagy PB, Cawley P. Study and comparison of different EMAT configurations for sh wave inspection. IEEE Trans Ultrason Ferroelectr Freq Control 20;58(2): [23] Chen Y, Kriegermeier-Sutton BK, Snyder JE, Dennis KW, McCallum RW, Jiles DC. Magnetomechanical effects under torsional strain in iron, cobalt and nickel. J Magn Magn Mater 200;236( 2):3 8. [24] Momeni Sepandarmaz, Konrad Adalbert, Sinclair Anthony N. Determination of reversible permeability in the presence of a strong bias field. J Appl Phys 2009;05: [25] Thompson RB. Mechanisms of electromagnetic generation and detection of ultrasonic lamb waves in iron-nickel alloy polycrystals. J Appl Phys 977;48(2): [26] steel08_permag.dat; 202. / [27] Chari MVK, Reece P. Magnetic field distribution in solid metallic structures in the vicinity of current carrying conductors, and associated eddy-current losses. IEEE Trans Power Appar Syst 974;PAS-93(): [28] Ribichini R, Cegla F, Nagy PB, Cawley P. Experimental and numerical evaluation of electromagnetic acoustic transducer performance on steel materials. NDT & E Int 202;45():32 8. [29] Scruby CB, Drain LE. Laser ultrasonics: techniques and applications. Taylor & Francis; 990. [30] Davies SJ, Edwards C, Taylor GS, Palmer SB. Laser-generated ultrasound: its properties, mechanisms and multifarious applications. J Phys D Appl Phys 993;26: