THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

468 16. UltrasonicsThe Physics of Magnetostrictive TransducersMagnetrostriction theory is highly analogous to that of piezoelectricity, but, in thiscase, account is taken of the polarizing field H 0 . We now consider a ferromagnticrod undergoing polarization throughout its length with a magnetic field H 0 , with B 0denoting the associated flux density. The resultant strain ε 0 is directly proportionalto the square of the flux density, i.e.,ε 0 = CB0 2 (16.35)where C is a constant. It is seen from Equation (16.35) that the sign of the resultantstrain is independent of the direction of the field. Now if we apply an excitingmagnetic field of strength H, which is appreciably less than H 0 with an associatedflux density B, we can writeB = μ i H = B 0 ≪ B 0 (16.36)where we have denoted μ i as the incremental magnetic permeability. For a stateof constant stress, we have for the resulting strain by differentiation of Equation(16.35):ε = ε 0ε = 2CB 0 B 0orε = 2CB 0 B = βμ i H (16.37)Here β = 2CB 0 constitutes the magnetostrictive strain coefficient (given in units ofm 2 /weber) that applies to small strains. Equation (16.36) is analogous to Equation(16.22) for no-load conditions. When no alternating field is applied, the value ofthe strain ε is given by Hooke’s law,ε = σ/Y (16.38)in terms of stress σ and Young’s modulus Y . We can now obtain the analog toEquation (16.23) by using H instead of the electric field E, Y instead of 1/a, andβμ i instead of d. This gives usε = σ Y + βμ i H (16.39)for a rod undergoing simultaneously a tensile stress σ and a magnetic field H. Theanalogy extended to Equation (16.20) yieldsB = σβμ i + μ i H (16.40)Clamping the rod causes strain ε to be zero in Equation (16.38), yieldingσ = Yβμ i H = Bwhere σ denotes the compressive stress and = Yβ represents the magnetostrictivestress constant (given in units of newton/weber). The reciprocal of , preferredby some authors, is called the piezomagnetic constant (units of weber/newton).