Discussion: “An Energy Method for Analyzing Magnetoelastic Buckling and Bending of Ferromagnetic Plates in Static Magnetic Fields” (Yang, W., Pan, H., Zheng, D., and Cai, Z., 1999, ASME J. Appl. Mech., 66, pp. 913–917)

The authors gave an energy method to analyze the magnetoelastic buckling and bending of ferromagnetic plates in different static magnetic fields. The elastic strain energy of Eq. (2) employed in this paper is for the bending of the beam-type plate. And in the derivation of magnetic energy of Eq. (5), the effect of end edges on magnetic fields is not taken into account. After the longitudinal and transverse demagnetizing factor $Nl$ and $Nh$ are calculated by Eqs. (12)–(13), respectively, the expressions of critical field $Bcr$ and bending deformation δ at free end are formulated by Eqs. (14) and (17), respectively. In this approach, the effect of width, denoted by w here, is considered only in the demagnetizing factors but not in the deformation. If a rectangular ferromagnetic plate under consideration is constrained by simple or clamped supports along the edges normal to the direction of width, it is possible that the same results for the magnetoelastic interaction will be obtained since $Nl$ and $Nh$ are independent on the boundary conditions. In other words, the results given in this paper are independent upon the support conditions of the edges along the longitudinal direction, which is obviously in contradiction to the practical problems. When the width of a rectangular plate increases to infinite, from the theory of plates, we know that the deflection of the plate approaches to that of a corresponding beam-type plate. When χ is very large, e.g., $103$ order in Moon and Pao 1 to this case, the condition $1/χ≪Nl=1$ is satisfied $(Nl=1$ may be got by Eq. (12) when $w→∞$⁠). According to Eq. (14a), however, it is found that the critical magnetic field $B¯cr$ for this case of cantilevered plates in transverse magnetic fields approaches to infinite. This results in contradiction to the finite critical magnetic fields given in literature to the same problem, e.g., Moon and Pao 1, Zhou et al. 2, and Zhou and Zheng 3 which are in agreement with the experimental data (1,4). For the prediction of bending of the plate in this paper, it is found by Eq. (17) and Fig. 3 that the incident angle α of the magnetic field does not influence the critical magnetic field $Bcr$ of the magnetoelastic instability. This result is also in contradiction to the conclusion given in the literature using the imperfect sensitive analysis in Popelar 5 and the numerical analysis in Zhou et al. 2. In fact, both the experimental measurement (1,6) and theoretical research display a fact that the critical magnetic field of a cantilevered ferromagnetic plate in transverse magnetic field is sensitive to the imperfect of incident angle of misalignment or oblique magnetic field. That is one of reasons why the theoretical predictions for the perfect case of the cantilevered plate in transverse magnetic field (1,3,7 for example) are almost higher than their experimental data (2). For the case of a ferromagnetic plate in longitudinal magnetic field, the authors gave a differential Eq. (20) which indicates that there is neither bend nor buckle. The authors did not give a comparison of their theoretical prediction and the experimental data to the increasing of natural frequency of the considered plate (8). Zhou and Miya 9 successfully gave a theoretical prediction of this problem. For the general model of magnetoelastic interaction for ferromagnetic plate structures and bodies in arbitrary magnetic fields, by which the experimental phenomena of magnetoelastic buckling, bending and increasing of natural frequency can be described, it can be found in Zhou and Zheng 10 11. It is obvious that these recent researches of magnetoelastic interaction do not support the opinion of authors: “It seems that no further progress has been made in theoretical analysis since the Moon-Pao theory was presented.”