Equation of state for ferromagnetic transducers Smart Structures and Materials 2002: Active Materials: Behavior and Mechanics Christopher S. Lynch, Editor, Proceedings of SPIE Vol. 4699, 463-472 (2002) Carl S. Schneider* Physics Department, U.S. Naval Academy, Annapolis, MD USA 21402 ABSTRACT An equation of state based upon magnetization scaled to its saturation value, m = M/M s , applied magnetic field scaled to its coercive value, h = H/H c , distribution of coercivities, and reversible susceptibility X rev of magnetization and applied stress, scaled to its initial value is proposed for ferromagnetic transducers. Reversible susceptibility divided by the initial susceptibility is the anisotropy function of domain magnetization, decreasing for Terfenol-D nearly linearly with scaled magnetization from one in the demagnetized state to zero at saturation. Measurements of reversible susceptibility, initial, anhysteretic and saturate magnetization curves, and loops for Terfenol-D show that differential magnetic susceptibility is the product of the reversible susceptibility and a cooperative function due to domain interactions. This function is roughly triangular in magnetization having the same slope from each reversal for magnetization magnitude up to half of saturation, at which the onset and decay of cooperation occur. This cooperative function causes parabolic Rayleigh minor loops and sigmoid major B(H) curves truncated by stress demagnetization. Anhysteretic reluctivity increases linearly with the root sum square of shape and stress demagnetizations, the latter added to the anisotropy function. Magnetization loops under stress are modeled and the d* transducer constant dB/d  is derived as a function of stress and magnetization. Keywords: transducer, ferromagnetic, hysteresis, stress, magnetostriction, anisotropy, susceptibility, anhysteretic 1. BACKGROUND OF THE ANISOTROPIC COOPERATIVE MODEL The extension and force exerted by a ferromagnetic transducer 1 are nearly single valued functions of their state of magnetization, but quite hysteretic functions of the magnetic field applied by solenoid currents around the transducer. This is due to the hysteretic relationship between applied magnetic field and resulting magnetization in ferromagnetic materials. Several descriptions of hysteresis have evolved since the first parabolic or nonlinear law of Lord Rayleigh 2 over a century ago. The molecular field of Weiss 3 first explained the presence of magnetic domains of saturated magnetization shown by Bitter pattern microscopy. The shape of hysteresis curves represents not only variations in the strength of wall pinning throughout the domains, but also the effect of domain field interactions studied experimentally 4 , theoretically 5 and through computer simulations 6 . Our recent anisotropic cooperative model 7 analytically combines the susceptibility of domains to rotation and wall bowing, manifested through the reversible Fig. (1) The anisotropic cooperative model of ferromagnetic hysteresis accurately predicts virgin and saturate curves and minor loops for annealed magnets. *Email: schneide@usna.edu Orthonol B(T) . . . Data ____ Model Virgin, Saturate Reversals -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 -20 -10 0 10 20 H(A/m)