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     

平面圆形线圈的磁弹性屈曲

本文以 Tokamak 装置中的角向场线圈为背景,导出了平面圆形线圈在周向和横向磁场作用下的磁弹性屈曲条件。发展了 F.C.Moon 的结果。     

A theoretical model of magnetoelastic bucking for soft ferromagnetic thin plates

As an essential model of magnetoelastic interaction between magnetic field and mechanical deformation, the study on magnetoelastic buckling phenomenon of soft ferromagnetic plates in a magnetic environment has been conducted. One of the key steps for the theoretical prediction of the critical magnetic field is how to formulate magnetic force exerted on the magnetized medium. Till today, the theoretical predictions, from theoretical models in publications, of the magnetoelastic buckling of ferromagnetic cantilevered beam-plate in transverse magnetic field are all higher than their experimental data. Sometimes, the discrepancy between them is as high as 100%. In this paper, the macroscope formulation of the magnetic forces is strictly obtained from the microscope Amperion current model. After that, a new theoretical model is established to describe the magnetoelastic buckling phenomenon of ferromagnetic thin plates with geometrically nonlinear deformation in a nonuniform transverse magnetic field. The numerical method for quantitative analysis is employed by combining the finite elemental method for magnetic fields and the finite difference method for deformation of plates. The numerical results obtained from this new theoretical model show that the theoretical predictions of critical values of the buckling magnetic field for the ferromagnetic cantilevered beam-plate are in excellent agreement with their experimental data. By the way, the region of applicability to the Moon-Pao's model, or the couple model, is checked by quantitative results. This project was supported in part by the National Natural Science Foundation of China and the Foundation of the SEdC of China for Returned Chinese Scholars from Abroad.     

MULTI-FIELD COUPLING BEHAVIOR OF SIMPLY-SUPPORTED CONDUCTIVE PLATE UNDER THE CONDITION OF A TRANSVERSE STRONG IMPULSIVE MAGNETIC FIELD

In order to study the multi-field coupling mechanical behavior of the simply-supported conductive rectangular thin plate under the condition of an externally lateral strong impulsive magnetic field, that is the dynamic buckling phenomenon of the thin plates in the effect of the magnetic volume forces produced by the interaction between the eddy current and the magnetic fields, a FEM analysis program is developed to characterize the phenomena of magnetuelastic buckling and instability of the plates. The critical values of magnetic field for the three different initial vibrating modes are obtained, with a detailed discussion made on the effects of the length-thickness ratio a/h of the plate and the length-width ratio a/b as well as the impulse parameterτon the critical value BOcr of the applied magnetic field.     

Magnetostatic deformation experiment: Bending of a thin plate

The Euler buckling formula is a simple deduction from the Bernoulli-Euler theory of bending. Perhaps a correct magnetoelastic buckling formula with be obtained as a simple deduction from a correct theory of magnetoelastic bending. The present investigation is a first step toward creating such a theory.Thin rectangular plates of solft steel were rigidly fixed at one end and caused to bend into a cylindrical shape by applying a uniform magnetic field whose direction made a small angle (five degrees) with the normal to the undeformed plate surface. Strain measurements and field measurements were made along top and bottom surfaces of the plate. A bending-moment diagram was calculated from the strain measurements and a bending-moment diagram was calculated from field measurements, using the integrand of the first Maxwell stress formula to calculate the force distribution acting on the plate. To get agreement between the two diagrams it was necessary to add a concentrated couple and transverse force (not revealed by the field measurements) at the free end of the plate. The postulated couple and force alone accounted for approximately 30 percent of the bending moment values calculated from strains. We conclude that if the external magnetic field at the surface of a magnetic body cann be calculated correctly, the integrand of the Maxwell formula will corretly give the force distribution over the surface, and classical elasticity theory will then given correctly the elalstic deformation.Paper was presented at the 1985 SEM Spring Conference on Experimental Mechanics held in Las Vegas, NV on June 9–14.     

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.     

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.     

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.     

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.     

Magnetoelastic buckling of thin plates in a uniform transverse magnetic field

The buckling of thin soft ferromagnetic elastic plates in a uniform transverse magnetic field is discussed. Based on a linearized version of the general magnetoelastic equations, a plate equation is derived together with the pertinent boundary conditions. By searching a nontrivial solution of this homogeneous system, a buckling value can be found. As an example, the circular plate is considered. In this paper, Maxwell stresses are used, but it is shown that identical results can be obtained by using other models (especially Amperian current) for the magnetoelastic stresses.

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Zusammenfassung In dieser Arbeit wird das Knicken dünner ferromagnetischer, elastischer Platten in uniformen magnetischen Feldern behandelt. Die Ableitung einer Plattengleichung zusammen mit den bezüglichen Randesbedingungen wird auf die linearisierten Gleichungen für weiche ferromagnetische Medien begründet. Durch Lösung dieses Systems wird eine Knicklast ermittelt. Die Methode wird anhand des Beispieles der kreisförmigen Platte illustriert. In dieser Abhandlung werden Maxwell Spannungen verwendet, aber es wird gezeigt dasz andere Modelle (insonderheit Amperesche Stromstärke) für die magnetoelastischen Spannungen das gleiche Resultat bilden.

     

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。     

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.     

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。      

Magnetoelastic buckling of a rectangular block in plane strain

Of interest here is the stability of a rectangular block subjected to a uniform magnetic field perpendicular to its longitudinal axis. The two ends of the block are frictionless and kept parallel to each other. This boundary value problem is motivated by the classical problem of magnetoelastic buckling in which a cantilever beam subjected to a transverse magnetic field buckles when the applied field reaches a critical value.This work presents a finite strain continuum mechanics formulation of the stability problem of a homogeneous, compressible, magnetoelastic rectangular block in plane strain subjected to a uniform transverse magnetic field. The applied variational approach employs an unconstrained energy minimization recently proposed by the authors.The analytical solution for the critical buckling fields for both the antisymmetric and symmetric modes are obtained for three different constitutive laws. The corresponding result for thin beams is extracted asymptotically for a special material and the solution is compared to previously published results. The critical magnetic field is shown to increase monotonically with the block's aspect ratio for each material and mode type. Antisymmetric modes are always the critical buckling modes for stress saturated and neo-Hookean materials, except for a narrow range of moderate aspect ratios (about 0.25) where symmetric modes become critical. For strain-saturated solids no buckling is possible above a maximum aspect ratio.     

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.     

Stress–strain state of flexible ring plates of variable stiffness in a magnetic field

A nonlinear two-dimensional model of a magnetoelastic flexible current-carrying ring plate is developed.Asystem of nonlinear equations describing the stress–strain state of flexible current-carrying plates in nonstationary mechanical and electromagnetic fields is derived. The stress state of a flexible plate of variable stiffness in a magnetic field is determined     

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.     

Use of magnetorheological elastomer in an adaptive sandwich beam with conductive skins. Part I: Magnetoelastic loads in conductive skins

In this study, the magnetoelastic loads for a vibrating conductive beam exposed perpendicularly to an applied steady magnetic field are addressed analytically by considering the effect of finite dimensions. The dynamic equation of such vibrating beam is presented and a simply supported conductive beam is simulated. The simulation indicates that the magnetoelastic loads affect the dynamic properties of the vibrating beam significantly only when the thickness of the beam is extremely small. This work is the basis for investigating and analyzing the field-controllable dynamic properties of a sandwich beam composed of conductive thin outer skins and a magnetorheological elastomer core, which will be presented in the second part of this research.     

Nonlinear Responses of a Magnetoelastic Beam in a Step-Pulsed Magnetic Field

A theoretical study is carried out on the dynamics of a magnetoelastic beam being in a step-pulsed magnetic field. For this aim, the magnetic potential and elastic energies are determined for the beam and partial differential equations are established according to Hamilton's principle. It is proven that the magnetoelastic beam can give a variety of complex behavior in the case of step-pulsed field excitations. An intermediate regime of two-well chaos is observed. Theoretical findings were found to be in a good agreement with the experimental results for the specific system parameters. On leave from Institute of Physics, University of Bayreuth, 65440 Bayreuth, Germany An erratum to this article is available at .     

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.