IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 6, NOVEMBER 2004 3463 Simulation of Surface Heating Effects and Effective Permeabilities Using a Jiles–Atherton Model George Lloyd, Ming L. Wang, Varsha Singh, and Ernesto Indochochea Abstract—The hysteretic, nonlinear character of ferromag- netic steels, coupled with pronounced stress and temperature sensitivities, make detailed magnetoelastic sensor design and calibration very difficult at present, particularly under possible high repetition rates during disaster scenarios, and in large cables ( R Bi , where is the magnetic skin depth and Bi is the thermal Biot number) with inhomogeneous stress and thermal fields. The desire also exists to associate bulk magnetic material parameters with microstructural features to enable generally useful correlations to be developed. The Jiles–Atherton class of thermodynamic models potentially fit these requirements, and we explore their use for investigating field thermomagnetic effects in A36/1018 steels. Model development is supported by microstructural characterization of the anhysteretic via magnetic force microscopy and tunneling electron microscope imaging. Index Terms—Dislocations, magnetic force microscopy (MFM), magnetoelasticity, tunneling electron microscopy (TEM). I. JILES–ATHERTON MODEL T HE SUITE of equations which define the Jiles–Atherton (JA) model are the first-order differential equation for the irreversible susceptibility, and the reversible susceptibility [1] (1) The parameter represents the linearized reversible component of magnetization due to reversible wall bending and translation. The pinning parameter is proportional to the domain wall-pin- ning site density and pinning strength. The director indicates the direction of traversal of the applied field. Reference [2] has suggested a form of to include the effects of dislocation den- sity and grain spacing . In these expressions, is the effective field and is the bulk magne- tization (magnetic dipole density per unit volume). In the case Manuscript received October 16, 2003. This work was supported by the Na- tional Science Foundation under Grant NSF 0085204. This paper was presented at The Ninth Joint Magnetism and Magnetic Materials–International Magnetics Conference, Anaheim, CA, January 5–9, 2004. See IEEE Trans. Magn., vol. 40, July 2004, Part II. G. Lloyd, M. L. Wang, and E. Indochochea are with the Department of Civil and Materials Engineering, University of Illinois at Chicago, Chicago, IL 60607 USA (e-mail: lloydg@asme.org; mlwang@uic.edu; jeindaco@uic.edu). V. Singh was with the Department of Civil and Materials Engineering, Uni- versity of Illinois at Chicago, Chicago, IL 60607 USA. She is now with Smart Structures, Rantoul, IL 61866 USA (e-mail: varsha@smart-structures.com). Digital Object Identifier 10.1109/TMAG.2004.832274 of completely isotropic materials under stress, the anhysteretic magnetization is given by (2) The term is the Langevin function, is the first-order internal exchange energy coupling coefficient, and is the effect of a constant mechanical stress on the equivalent field with the magnetization, i.e., magnetoelastic coupling [3]. The parameter , a pseudodomain anhysteretic form factor or scaling factor for the effective field, here can be related to the Langevin model for a paramagnetic material. It is a function of temperature and domain density in the demagnetized state (3) where is Boltzmann’s constant, J/K, is the thermodynamic temperature, is the vacuum permeability, (N/A ), and (A m ) is the anhysteretic saturation magnetic dipole moment of a pseudodomain. Together, the pa- rameters and (and thus the temperature) dominate the slope of the resulting hysteresis curves. II. SAMPLE The material investigated was a Grade-60 hot-rolled steel (ASTM A36), used for shear reinforcement in concrete slabs and beams. This steel is classified as a hypoeutectoid low carbon steel (nominal composition C 0.15%, Si 0.16%, Mn 0.65%, P 0.028%, S 0.027%). The microstructure consists of approximately 83% proeutectoid ferrite and 17% pearlite (the former being the ferromagnetic phase). Ferrite is a solid solu- tion of carbon in iron, with a maximum solubility of 0.0218%; it has a body-centered crystal structure, and is mechanically and magnetically soft. Pearlite is an alternating lamellar struc- ture of mechanically hard cementite and ferrite; pearlite acts strongly as a pinning agent. Cementite has a carbon content of 6.67 wt.% C, and thus, is mechanically hard and brittle. The ferrite grain size is about 15 m, as shown in Fig. 1 (left). The magnetic properties were measured at several temperatures using equipment and procedures described previously [4]; The JA model was manually fit to the data commencing with values of close to measured values of . Domain imaging was performed in an attempt to relate the micromagnetic features to the micromagnetic parameters of the Jiles model. Imaging was conducted with a Digital Instruments 0018-9464/04$20.00 © 2004 IEEE