AIM CONFERENCE 2016 -NON DESTRUCTIVE EVALUATION AND TESTS
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On the experimental validation of a numerical model of a magnetic probe for material characterization M. d’Aquino1, S. Minucci2, C. Petrarca2, G. Rubinacci2, A. Tamburrino3,4, S. Ventre3 1
1Dipartimento di Ingegneria, Università di Napoli "Parthenope", Napoli, 80143 (Italy) 2 DIETI, Università di Napoli Federico II, 80125, Napoli, Italy 3 DIEI, Università di Cassino e del Lazio Meridionale, 03043, Cassino, Italy 4 ECE, Michigan State University, East Lansing, Michigan 48824, USA
[email protected]
Abstract—The contribution is aimed to describe the preliminary results obtained in the experimental validation of a 3D numerical model of a magnetic probe for material characterization. The model is fully three-dimensional and includes a 3D vector hysteresis constitutive relationship based on the Jiles-Atherton approach. Periodic voltage sources and eddy currents are efficiently modeled leading, after discretization, to a non linear dynamical system, efficiently solved in the frequency domain by means of a Picard-Banach iterative approach. Index Terms— nondestructive testing and evaluation, vector hysteresis, integral equations, finite element analysis, harmonic analysis.
I. INTRODUCTION Monitoring of early degradation in materials is a crucial issue in many fields of application. A possible approach is based on the concept that the material microstructure affects both the mechanical and the magnetic properties of ferromagnetic materials. In this way, the electromagnetic nondestructive evaluation methods achieve the characterization of mechanical properties by using a suitable combination of magnetic measurements such as the incremental permeability along a hysteresis loop, multifrequency eddy currents induced in the hysteretic medium, Barkhausen noise and harmonic analysis [1]. The validity of this approach has been already demonstrated by several studies and their application in industrial environments [1], [2]. A reliable interpretation of the measurements as well as an effective design of the inspection system can be efficiently carried out by means of accurate and robust 3D numerical models. Such models must be able to compute the field distribution induced by periodic voltage sources in a ferromagnetic specimen characterized by a 3D vector hysteresis model. In this paper, the results of the numerical model presented in [3], [4] will be compared with the experimental measurements, for a better understanding of the experimental protocol at the basis of the measurements of quantities related to the incremental permeability. Preliminary results have been already discussed in [4], [5]. II.
MATHEMATICAL MODEL
The magnetic probe (see Fig. 1) is made of a soft magnetic yoke above the ferromagnetic specimen to be tested. A set of low resistivity excitation coils driven by a low frequency
voltage V is wounded on the yoke. Another coil placed between the legs of the yoke is powered at high frequency. The impedance of this high frequency coil is in a close relationship with the incremental permeability at the working points of the main hysteresis loop determined by the low frequency excitation. The electromagnetic behavior of this system is described by the set of magneto-quasi-stationary Maxwell equations with suitable constitutive relationships, leading to the following weak form in terms of the unknown current density J and magnetization M: d ∫V w ⋅η JdV + dt V∫ w ⋅ A ( J, M ) + A s dV C C (1) NE + ∑ ∫ ϕ h w ⋅ nˆ dS = 0, ∀w ∈ S (Vc ) h =1 S h
∫ u ⋅G ( M ) dV= ∫ u ⋅ B ( J, M ) + B -1
VM
s
2 dV , ∀u ∈ L (VM )
(2)
VM
where S = {J ∈ H (div, VC ), ∇ ⋅ J = 0 in VC , J ⋅ nˆ = 0 on ∂VC \ S E } , A s and Bs are the magnetic vector potential and the flux density fields generated by the external sources and the (linear)
operators A and B provide the same quantities for prescribed magnetization and current density, respectively. NE electrodes described by the surface S E = ∪h =1:N E S h may be present as a part of the boundary ∂VC of the conducting domain. The electric potential φh is assigned on each electrode. The conducting material is described by E = η J in VC. The magnetic material is characterized by the constitutive equation M = G ( B ) in VM, an isotropic vector generalization of the
classical Jiles-Atherton model. This is a complex constitutive relationship that is expressed in terms of differential forms (see [3, 4] and references therein). The two time scales determined by the two sources powered at different frequencies can be decoupled in two sub-problems (by linearization) if the field due to the second source is weak enough. The first problem (computing the field along the main hysteresis loop) can be solved in the frequency domain by a fixed point procedure [6] where great efficiency is achieved by expanding the unknowns by (few) Fourier harmonics [3]. The second problem (computing the response of the small coil) can
AIM CONFERENCE 2016 -NON DESTRUCTIVE EVALUATION AND TESTS
be computed by solving a proper anisotropic linear problem [4].
2 with the experiment.
Fig. 1. The configuration of the magnetic probe.
Fig. 2. The current experimental setup. III.
EXPERIMENTAL SET-UP AND PRELIMINARY RESULTS
A. Experimental set-up Due to the inherent three-dimensional behavior of the magnetic probe here described, we decided to make a preliminary validation of the numerical model, by imposing the field due to the high frequency source in the same direction of the main low frequency field. In this way, the consistency of the 3D vector hysteresis model can be assessed with reference to a field geometry that is simpler so that also the limits of the linearization can be more easily assessed. The experimental set-up is shown in Fig. 2. Its main parameters are summarized in Table I TABLE I PARAMETERS OF THE EXPERIMENTAL SET-UP Parameter Value Parameter Value D1=7.5 cm Turns of LF coils 190 - 35 LF coils length D2=1.8 cm Turns of HF coil 100 HF coil length D3=6.0 cm fLF 10Hz fHF 5kHz h1=9.1 cm, δ=1.1 mm Strip L1=12.6 cm yoke dimensions Sp=2 cm dimensions 2 S=2.8×2.0 cm RLF - Rcable 140mΩ - 20mΩ RHF 4.8 Ω
B. Preliminary results As a first test, we compared the flux measured through the legs of the yoke and through the ferromagnetic strip, respectively, with the fluxes computed at the same locations by the numerical model. The results shown in Fig. 3 seem to be very satisfactory. In the full paper we will detail the model used for simulating the incremental permeability, within the Jiles-Atherton vector hysteresis model and the comparison
Fig. 3. Top: The magnetic flux linked with the coil wounded on the ferromagnetic strip. Bottom: The magnetic flux linked with the coil wounded on one of the leg of the yoke.
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[2]
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[7]
D. C. Jiles, Magnetic methods in nondestructive testing, in Encyclopedia of Materials Science and Technology, K. H. J. Buschow, Ed. Oxford, U.K., Elsevier, 2001. S. Takahashi, S. Kobayashi, Y. Kamada, H. Kikuchi, K. Ara, NDE Method Using Minor Hysteresis Loops …., in Electromagnetic Nondestructive Evaluation (X), S. Takahashi et al. Eds., Ios Press (2007) 209–216. M. d'Aquino, G. Rubinacci, A. Tamburrino, S. Ventre, Efficient Numerical Solution of Magnetic Field Problems in Presence of Hysteretic Media for Nondestructive Evaluation, IEEE Trans. Magn., vol. 49, no. 7, pp. 3167–3170. 2013 M. d'Aquino, G. Rubinacci, A. Tamburrino, S. Ventre, ThreeDimensional Computation of Magnetic Fields in Hysteretic Media with Time-Periodic Sources, IEEE Trans. Magn. vol. 50, no. 2, pp. 53–56, 2014. M. d’Aquino, S. Minucci, C. Petrarca, G. Rubinacci, A. Tamburrino, S. Ventre, A 3D numerical model with vector hysteresis for modelling the electromagnetic inspection of ferromagnetic materials, The 17th Int. Symposium on Applied Electromagnetics and Mechanics (ISEM2015) 15- 18 September 2015, Kobe, Japan R. Albanese, F. I. Hantila, G. Rubinacci, A nonlinear eddy current integral formulation in terms of a two-component current density vector potential, IEEE Trans. Magn., vol. 32, no. 3, pp 784-787, 1996. H.-E. Chen, S. Xie, H. Zhou, Z. Chen, T. Uchimoto, T. Takagi, Y. Kensuke, Numerical simulation of magnetic incremental permeability for …, Int. J. of App. Electromagnetics and Mechanics 45 (2014) 379– 386.
