软铁磁薄板磁弹性屈曲的理论模型

铁磁弹性薄板的磁弹性屈曲问题一直作为电磁——弹性力学相互作用的一个基本模型进行研究，而作用在其磁介质上的磁力计算则是定量理论预测准确与否的关键．到目前为止，文献上已有的理论模型对悬臂铁磁梁式悬臂板在横向磁场中磁弹性屈曲的理论预测值始终高于实验值，有的甚至相差１００％左右．本文基于电磁力计算的微观安培电流模型，严格给出了软铁磁薄板等效横向磁力的宏观计算表达式．在此基础上，建立了电磁——力学相互耦合作用的非线性理论模型．该模型能描述铁磁薄板结构在非均匀横向磁场环境中的磁弹性失稳（或屈曲）．其定量分析采用了有限元法和有限差分法相结合．数值结果显示：本模型给出的磁弹性屈曲的临界磁场值与实验值符合良好．与此同时，文中还对文献中认为较成功的Ｍｏｏｎ－Ｐａｏ模型的基本假设进行了分析．定量结果发现：Ｍｏｏｎ－Ｐａｏ理论模型的基本假设仅在梁式板的长厚比Ｌ／ｈ比较大时（约在２００左右），是可以接受的，而当Ｌ／ｈ较小时，该假设将导致理论值与实验值的较大误差．Ｌ／ｈ比值越小，理论值与实验值的误差越大     

An analysis on characteristic of magnetoelasticity of a magnetic field microsensor made of cantilevered plates with ferromagnetic films

The purpose of this paper is to study the magnetomechanical characteristic of a microsensor which is composed of a cantilevered beam-plate with ferromagnetic films in order to measure the magnetic field from the deformation of plate when the microsensor is located in the magnetic field. To this end, a numerical approach made up of the finite element method for magnetic field and the finite difference method for deflection of the microsensor is proposed to perform the numerical analysis of deflection under magnetoelastic interaction. Some quantitative results of a case study for the magnetoelastic characteristic between the magnetic field and deflection of the microsensor in the magnetic field are given. The results show that this microsensor can be used not only to measure the magnitude of magnetic intensity, but also to possibly monitor the direction of the vector of the magnetic field. The project supported by the NNSFC(No. 19772014), the China National Foundation for Outstanding Young Researchers(No. 19725207), and Foundation for University Key Teacher by the Ministry of Education of China.     

铁磁梁式板磁弹性初始后屈曲及缺陷敏感性分析

从铁磁板的磁弹性广义变分原理出发，通过对铁磁板内外的磁场和板变形 场的摄动技术以及采用经典Koiter理论，对横向磁场中铁磁悬臂、简支、固支梁式薄 板磁弹性稳定性的初始后屈曲行为及缺陷敏感性进行了定性研究. 解析地给出了悬臂板对初 始缺陷敏感，简支和固支板对初始缺陷不敏感等结果，对悬臂梁式板理论预测的临界磁场值 往往大于实验观测结果的现象从定性上给予了合理解释.     

考虑尺寸效应板梁的磁弹性屈曲

研究表明,有限尺寸效应是影响板——梁磁弹性稳定性分析精度的主要原因。本文用边界元法分析了板边沿的磁感强度集中,然后用里兹法计算了软铁磁材料悬臂板——梁的临界屈曲磁场。通过和Moon-Pao的理论和实验结果以及Miga的有限元结果比较,表明本方法具有良好的精度,且计算方法简便。     

Buckling and post-buckling of a ferromagnetic beam-plate induced by magnetoelastic interactions

Buckling and post-buckling behavior of a soft ferromagnetic beam-plate with unmovable simple supports under an applied magnetic field is analyzed by taking into account the non-linear effect of large deflections based on the von Karman model of plates. The study shows that buckling is proceeded by bending only if the applied magnetic field forms an incident angle with the normal of the plate. The characteristics of buckling and post-buckling are explored by numerical analysis. In particular, the numerical solution shows that, when the applied magnetic field is oblique, the plate snapps from a whole wave configuration into a half-wave configuration as buckling/snapping occurs, and that the magnetoelastic buckling strength increases with the increasing oblique angle.     

悬臂铁磁板磁弹性耦合作用的力学分析

对有限板宽的悬臂铁磁板在外磁场中的磁弹性相互耦合作用的力学行为，建立了描述板弯曲和稳定性的理论模型及有限元定量分析程序，研究了外磁场倾斜角对板磁弹性失稳的临界磁场值的影响。     

磁弹性簿板屈曲的研究进展和存在的若干问题

着重评述了近２０年来有关铁磁材料弹性薄板处在均匀横向外加磁场中发生屈曲这一新的磁－力耦合问题的理论与实验研究进展，在此基础上，介绍了笔者近年来围绕这一问题所开展的工作以及尚需进一步研究的几个问题。     

磁弹性簿板屈曲的研究进展和存在的若干问题

着重评述了近２０年来有关铁磁材料弹性薄板处在均匀横向外加磁场中发生屈曲这一新的磁－力耦合问题的理论与实验研究进展,在此基础上,介绍了笔者近年来围绕这一问题所开展的工作以及尚需进一步研究的几个问题。      

斜磁场作用下铁磁梁式板的磁场摄动分析

基于磁场摄动技术和广义变分原理得到的磁力公式,对铁磁梁式板在斜磁场作用下的磁场和磁力摄动展开研究.给出了考虑磁场端部效应情形时铁磁梁式板的摄动磁场及其磁力摄动表达式,使得解析分析斜磁场作用下铁磁梁式板磁弹性耦合问题成为可能.研究表明:只有考虑了磁场端部效应的斜磁场磁力摄动公式才能模拟铁磁结构的弯曲模式且能够定性揭示铁磁简支梁式板弯曲构形为双半波形.     

Changes in the natural frequency of a ferromagnetic rod in a magnetic field due to magnetoelastic interaction

Based on the magnetoelastic generalized variational principle and Hamilton＇s principle, a dynamic theoretical model characterizing the magnetoelastic interaction of a soft ferromagnetic medium in an applied magnetic field is developed in this paper. From the variational manipulation of magnetic scale potential and elastic displacement, all the fundamental equations for the magnetic field and mechanical deformation, as well as the magnetic body force and magnetic traction for describing magnetoelastic interaction are derived. The theoretical model is applied to a ferromagnetic rod vibrating in an applied magnetic field using a perturbation technique and the Galerkin method. The results show that the magnetic field will change the natural frequencies of the ferromagnetic rod by causing a decrease with the bending motion along the applied magnetic field where the magnetoelastic buckling will take place, and by causing an increase when the bending motion of the rod is perpendicular to the field. The prediction by the mode presented in this paper qualitatively agrees with the natural frequency changes of the ferromagnetic rod observed in the experiment.     

带铁磁薄膜悬臂板的磁场微感应器磁弹性特征研究

对于在可变形非磁材料悬臂梁式板单表面粘贴可磁化材料薄膜所构成的磁场微传感器件结构,研究了其处在磁场环境中的磁弹性弯曲变形的磁场-力学特征。为此,建立了由有限元方法分析磁场与有限差分法计算挠曲变形相结合来计算其结构在磁场中产生磁弹性变形的定量分析程序。在此基础上,对于这一微传感结构的算例给出了其结构变形随外加磁场环境变化的磁场-挠度特征关系等定量结果。结果表明：微传感器件不仅可以测量出磁场的大小,而且给出了测量磁场矢量方向的可能性。     

Magnetoelastic buckling: Why theory and experiment disagree

This paper does not stand alone; it is directly related to N.S. Christopherson's experimental study^1,2 of the magnetoelastic bending of thin steel plates, presented at a recent SEM meeting. It is, in fact, an extension of that study and relies upon some of Christopherson's data.In 1968, Moon and Pao presented a theory of the magnetoelastic buckling of a beam plate in a uniform magnetic field which differed from experimental results by a factor of two. Attempts to explain the discrepancy still leave approximately 25-percent error in the theoretical results as compared with experiment.We show that the assumption made by Moon and Pao—that a plate element experiences a force system consisting only of a couple, whose magnitude is proportional to the rotation of the element—is invalid for finite plates. An experiment is suggested for determining whether the assumption is valid for infinite plates. The present state of knowledge concerning magnetoelastic buckling is briefly summarized. Several types of magnetoelastic buckling are identified and briefly contrasted. Attention is directed to the importance of field discontinuities in problems involving magnetic deformation. Two paradoxical aspects of the Moon-Pao formula for the magnetoelastic buckling of beam plates are pointed out and briefly discussed.     

Magnetoelastic bending and stability of soft ferromagnetic rectangular plates

I.IntroductionSofar,twokindsofphenomenaofmagnetoelasticinteractiontoferromagneticplatesinappliedmagneticfieldshavebeenfound.Oneisthemagnetoelasticbuchlingorillstabllitytoacantileveredbeam-plateinatransversemagneticfieldt'],thatisthephen1Onlenollot'negiltivemagneticstiffness,whiletheotheristheincreaseofnaturalfrequencyoftheplatewhichisfreelyviratinginanin-planemagneticfieldl=],thatisthephenomenonofpositivemagneticstiffness.Inordertosimulatethesetwokindsofexperimentalphenomena,sometheoreticalmo…     

ANALYSIS ON THE MAGNETO-ELASTIC-PLASTIC BUCKLING/SNAPPING OF CANTILEVER RECTANGULAR FERROMAGNETIC PLATES

An analysis of buckling/snapping and bending behaviors of magneto-elastic-plastic interaction and coupling for cantilever rectangular soft ferromagnetic plates is presented.Based on the expression of magnetic force from the variational principle of ferromagnetic plates,the buckling and bending theory of thin plates,the Mises yield criterion and the increment theory for plastic deformation,we establish a numerical code to quantitatively simulate the behaviors of the nonlinearly multi-fields coupling problems by the finite element method.Along with the phenom- ena of buckling/snapping and bending,or the characteristic curve of deflection versus magnitude of applied magnetic fields being numerically displayed,the critical loads of buckling/snapping, and the influences of plastic deformation and the width of plate on these critical loads,the plastic regions expanding with the magnitude of applied magnetic field,as well as the evolvement of deflection configuration of the plate are numerically obtained in a case study.     

Magnetoelastic interactions in a soft ferromagnetic body with a nonlinear law of magnetization: Some applications

Linearized equations and boundary conditions of a magnetoelastic ferromagnetic body are obtained with the nonlinear law of magnetization. Magnetoelastic interactions in a multi-domain ferromagnetic materials are considered for magneto soft materials, i.e. the case when the magnetic field intensity vector and magnetization vector are parallel. As a special case, the following two problems are considered: (1) the magnetoelastic stability of a ferromagnetic plate-strip in a homogeneous transverse magnetic field; (2) the stress–strain state of a ferromagnetic plane with a moving crack in a transverse magnetic field. It is shown that the modeling of magnetoelastic equations with a nonlinear law of magnetization provides qualitative and quantitative predictions on physical quantities including critical loads and stresses. In particular, it is shown that the critical magnetic field in plate stability problems found with the nonlinear law of magnetization is in better agreement with the experimental finding than the one found with a linear law. Furthermore, it is also shown that the stress concentration factor around a crack predicted with the nonlinear law of magnetization is more accurate than the one obtained with a linear counterpart. Numerical results are presented for above mentioned two problems and for various forms of nonlinear laws of magnetization.     

Buckling and post-buckling analysis for magneto-elastic–plastic ferromagnetic beam-plates with unmovable simple supports

This paper presents an analysis of buckling/snapping, bending and post-buckling/snapping behaviors of magneto-elastic–plastic interaction and coupling for soft ferromagnetic beam-plates with geometrically nonlinear deformation and unmovable simple supports at the ends of the plates. Based on the expression of magnetic force from the variational principle of ferromagnetic plates, the theory of thin plates with the nonlinear deformation of van Karman’s type, and the Mises yield criterion and the increment theory for plastic deformation, here, we establish a numerical code to quantitatively simulate the behaviors of the nonlinearly multi-coupling problems by the finite element method. Along with that the phenomena of buckling, bending, and post-buckling/snapping, or the characteristic curve of deflection versus magnitude of applied magnetic fields are numerically displayed, the critical values of buckling/snapping and yield magnetic fields, and the expansibility of plastic region after the plates undergo plastic deformation with increasing of the applied magnetic fields, as well as the evolvement of deflection configuration of the plate are numerically obtained in a case study.     

Magnetoelastic buckling: Theory vs. experiment

Existing theoretical values and experimental values of magnetoelastic critical buckling fields for ferromagnetic beams and plates differ by a factor of two. The usual theoretical treatment assumes that the plate (a) is immersed in an infinite uniform field, (b) is infinitely wide and long, and (c) is made of a linearly magnetic material. We present an experimental investigation, combined with a simple analysis, which determines specific criteria which must be satisfied by experimental programs in order to validate the three assumptions. It is shown that experimental arrangements which validate (a) and (c) are feasible, but not (b). Hence, there is need for improved theoretical treatments which take into account the finite size of the specimens.     

Analytical and numerical studies of free vibration frequencies of ferromagnetic plates with arbitrary electric conductance in a transverse magnetic field in the 3D setting

The problem on the vibrations of a magnetoelastic ferromagnetic plate was studied in [1–5] from the viewpoint of the averaging approach, i.e., on the basis of the classical Kirchhoff hypothesis. In [6–10], a new approach proposed for elastic plates in [11] was used to derive dispersion relations for magnetoelastic plates. In [10], the 3D approach was used to obtain the ferromagnetic plate vibration frequencies; in the case of a transverse magnetic field, the equations of the perturbed motion of the plates were written out with the initial stresses taken into account [5, 12] but without considering the initial strains. In [13, 14], the problems on the vibrations of conducting plates in a magnetic field were studied.In the present paper, we derive the dispersion equations, which are asymptotic equations for small magnetic fields and exact equations for the initial stresses and strains related by Hooke’s law. The corresponding numerical computations are also performed.     

A magnetoelastic theoretical model for soft ferromagnetic shell in magnetic field

With the aid of a generalized variational method, in this paper, a theoretical model for soft ferromagnetic shells is derived to describe their magnetoelastic behavior in an applied magnetic field. Having made a quantitative comparison between the numerical predictions given by several theoretical models and the experimental results on strains of a cylindrical shell, we find that the predictions got by our model are in good agreement with the experimental data. It is also found that the Moon’s model is a special case of the model derived in this paper when the relative magnetic permeability μ_r>10⁴, which confirms that it is reasonable for the Moon’s model to calculate strains of the soft ferromagnetic shells. Having displayed the distribution of the equivalent magnetic force in the length of the cylindrical shell and its circumferential bending strains with different elastic end constraints, we give an explanation for the discrepancy between Moon’s analytical results and his experimental ones.