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.     

铁磁薄板在磁场中变形和应力的理论分析与实验研究

在二维板壳磁弹性理论基本方程基础上,以二维铁磁薄板磁弹性问题为例,建立了含有10个基本未知量的偏微分方程组,再利用Newmark有限等差式,得到了可以应用DOM方法求解的标准型方程组.求解得到了载流铁磁薄板的位移及应力与各电磁量之间的关系,计算了载流铁板在磁场中的变形和应力.同时进行了载流铁磁性薄板在电磁场中变形和应力的实验研究,介绍了电磁场中铁磁性薄板的实验装置和实验方法,给出了实验数据,并将实验结果与理论计算结果进行了分析对比.     

A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.     

铁磁导电梁式析在横向均匀磁场中的动力特性分析

对横向磁场中自由振动的导电铁磁梁式板,给出了考虑其磁化和涡电流与力学效应耦合影响的定解问题的基本方程。在对不同支承情况下板的振动频率和临界磁场值进行定量分析的基础上,讨论了电导率、磁化率和板厚等参数以及支承条件对板结构的振动频率与磁弹性临界磁场的影响。     

Study on generalized magneto-thermoelastic problems by FEM in time domain

This paper presents an investigation of temperature, displacement, stress, and induced magnetic field in a half space perfectly-conductive plate. Finite element equations regarding generalized magneto-thermoelasticity problems with two relaxation times (i.e., the G-L theory) are derived using the principle of virtual work. For avoiding numerical complication involved in inverse Laplace and Fourier transformation and low precision thereof, the equations are solved directly in time-domain. As a numerical example, the derived equation is used to investigate the generalized magneto-thermoelastic behavior of a semi-infinite plate under magnetic field and subjecting to a thermal shock loading. The results demonstrate that FEM can faithfully predict the deformation of the plate and the induced magnetic field, and most importantly can reveal the sophisticated second sound effect of heat conduction in two-dimensional generalized thermoelastic solids, which is usually difficult to model by routine transformation methods. A peak can be observed in the distribution of stress and induced magnetic field at the heat wave front and the magnitude of the peak decreases with time, which can not be obtained by transformation methods. The new method can also be used to study generalized piezo-thermoelastic problems.The project supported by the National Natural Science Foundation of China (10132010 and 10472089)The English text was polished by Yunming Chen     

Study on magneto–elastic–plastic deformation characteristics of ferromagnetic rectangular plate with simple supports

In this paper, the magnetic-elastic-plastic deformation behavior is studied for a ferromagnetic plate with simple supports. The perturbation formula of magnetic force is first derived based on the perturbation technique, and is then applied to the analysis of deformation characteristics with emphasis laid on the analyses of modes, symmetry of deformation and influences of incident angle of applied magnetic field on the plate deformation. The theoretical analyses offer explanations why the configuration offer- romagnetic rectangular plate with simple supports under an oblique magnetic field is in-wavy type along the x-direction, and why the largest deformation of the ferromagnetic plate occurs at the incident angle of 45°for the magnetic field. A numerical code based on the finite element method is developed to simulate quantitatively behaviors of the nonlinearly coupled multi-field problem. Some characteristic curves are plotted to illustrate the magneto--elastic-plastic deflections, and to reveal how the deflections can be influenced by the incident angle of applied magnetic field. The deformation characteristics obtained from the numerical simulations are found in good agreement with the theoretical analyses.     

Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory

In this research, thermal buckling of circular plates compose of functionally graded material (FGM) is considered. Equilibrium and stability equations of a FGM circular plate under thermal loads are derived, based on the higher order shear deformation plate theory (3rd order plate theory). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. A buckling analysis of a functionally graded circular plate (FGCP) under various types of thermal loads is carried out and the result are given in closed-form solutions. The results are compared with the critical buckling temperature obtained for FGCP based on first order (1st order plate theory) and classical plate theory (0 order plate theory) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of FGCP, whereas the first order and classical plate theory overestimates buckling temperature.     

Transient free convection of non-Newtonian fluids along a wavy vertical plate including the magnetic field effect

A Prandtl transformation method is applied to study the transient free convection of non-Newtonian fluids along a wavy vertical plate in the presence of a magnetic field. A simple transformation is proposed to transform the governing equations into the boundary-layer equations and solved numerically by the cubic spline approximation. A simple coordinate transformation is employed to transform the complex wavy surface to a vertical flat plate for a constant wall temperature by the numerical method. The effects of the magnetic field parameter, the wavy geometry and the non-Newtonian nature of the fluids on the flow characteristics and heat transfer are discussed in detail.     

Principal parametric resonance of axially accelerating rectangular thin plate in magnetic field

Nonlinear parametric vibration and stability is investigated for an axially accelerating rectangular thin plate subjected to parametric excitations resulting from the axial time-varying tension and axial time-varying speed in the magnetic field. Consid- ering geometric nonlinearity, based on the expressions of total kinetic energy, potential energy, and electromagnetic force, the nonlinear magneto-elastic vibration equations of axially moving rectangular thin plate are derived by using the Hamilton principle. Based on displacement mode hypothesis, by using the Galerkin method, the nonlinear para- metric oscillation equation of the axially moving rectangular thin plate with four simply supported edges in the transverse magnetic field is obtained. The nonlinear principal parametric resonance amplitude-frequency equation is further derived by means of the multiple-scale method. The stability of the steady-state solution is also discussed, and the critical condition of stability is determined. As numerical examples for an axially moving rectangular thin plate, the influences of the detuning parameter, axial speed, axial tension, and magnetic induction intensity on the principal parametric resonance behavior are investigated.     

轴向运动导电板磁弹性非线性动力学及分岔特性

研究磁场环境中轴向运动导电薄板磁弹性动力学及分岔特性。考虑几何非线性因素,在给出薄板运动的动能、应变能及外力虚功的基础上,应用哈密顿变分原理,得到磁场中轴向运动薄板的非线性磁弹性振动方程,并给出洛伦兹电磁力的确定形式。针对横向磁场环境中条形板共振特性进行分析,应用多尺度法和奇异性理论,得到稳态运动下的分岔响应方程以及普适开折对应的转迁集。通过算例,分别得到以磁感应强度、轴向运动速度和激励力为分岔控制参数的分岔图、最大李雅普诺夫指数图和庞加莱映射图等计算结果,讨论不同分岔参数对系统呈现的倍周期和混沌运动的影响。结果表明,通过相应参数的改变可实现对系统复杂动力学行为的控制。     

On the thermodynamics and stability of deformable dielectrics

A brief derivation of the equations for deformable dielectrics is presented. The basic theory generalises that due to Toupin [1], for it contains thermomechanical interactions and viscous dissipation within a non-equilibrium thermodynamic framework recently constructed by Bree and Beevers [2]. Some asymptotic and Lyapounov stability results are established for initial-boundary value problems in the linearised theory using extended energy arguments. Similar arguments are then applied to produce analogous stability results for small electro-mechanical perturbations superimposed on a large static deformation and a strong magnetic field.     

Combined heat and mass transfer by mixed convection MHD flow along a porous plate with chemical reaction in presence of heat source

An exact and a numerical solutions to the problem of a steady mixed convective MHD flow of an incompressible viscous electrically conducting fluid past an infinite vertical porous plate with combined heat and mass transfer are presented.A uniform magnetic field is assumed to be applied transversely to the direction of the flow with the consideration of the induced magnetic field with viscous and magnetic dissipations of energy.The porous plate is subjected to a constant suction velocity as well as a uniform mixed stream velocity.The governing equations are solved by the perturbation technique and a numerical method.The analytical expressions for the velocity field,the temperature field,the induced magnetic field,the skin-friction,and the rate of heat transfer at the plate are obtained.The numerical results are demonstrated graphically for various values of the parameters involved in the problem.The effects of the Hartmann number,the chemical reaction parameter,the magnetic Prandtl number,and the other parameters involved in the velocity field,the temperature field,the concentration field,and the induced magnetic field from the plate to the fluid are discussed.An increase in the heat source/sink or the Eckert number is found to strongly enhance the fluid velocity values.The induced magnetic field along the x-direction increases with the increase in the Hartmann number,the magnetic Prandtl number,the heat source/sink,and the viscous dissipation.It is found that the flow velocity,the fluid temperature,and the induced magnetic field decrease with the increase in the destructive chemical reaction.Applications of the study arise in the thermal plasma reactor modelling,the electromagnetic induction,the magnetohydrodynamic transport phenomena in chromatographic systems,and the magnetic field control of materials processing.     

铁磁矩形简支板的磁弹塑性弯曲行为分析

基于广义变分原理得到的磁力计算公式,采用塑性增量理论,Mises屈服准则和有效的增量有限元计算方法,研究了线性强化材料铁磁矩形板的磁弹塑性弯曲行为。在文中定量模拟了铁磁简支矩形板在外加磁场作用下的挠度特征曲线,铁磁板发生塑性变形时的构型图和不同外加磁场下的中截面构型,以及铁磁板在卸载后的残余挠度特征曲线等力学特征,分析了塑性区域随磁场增加而扩展的情况。数值结果表明:当铁磁矩形板上的部分区域发生塑性屈服后,其变形明显大于相同磁场条件下铁磁板发生的弹性变形值;且随着外加磁场倾角的增大(0°<α≤45°),铁磁板进入塑性屈服状态的临界屈服磁场值减小;铁磁板的中截面构形为双半波型,其塑性区域由铁磁板两侧挠度最大的区域向板的中心区域扩展,板的中心最后进入塑性区域等。     

ANALYSIS ON DYNAMIC CHARACTERISTICS FOR FERROMAGNETIC CONDUCTING PLATES IN A TRANSVERSE UNIFORM MAGNETIC FIELD

In this paper, a set of basic equations for free vibration of ferromagnetic conducting plates in a transverse magnetic field are presented, in which the coupled effects of magnetization and eddy current on the mechanical behavior of the plate are included. Based on the quantitative analyses on the vibration frequency and the values of the critical magnetic field for several supporting conditions of the plate, the effects of the conductivity, the magnetic permeability, the thickness of the plate and supporting conditions on the vibration frequency of the plate and the critical magnetic field are discussed. Supported by Outstanding Young Scientist Fund of NSFC(No. 19725207).     

A theory of finite deformation magneto-viscoelasticity

This paper deals with the mathematical modelling of large strain magneto-viscoelastic deformations. Energy dissipation is assumed to occur both due to the mechanical viscoelastic effects as well as the resistance offered by the material to magnetisation. Existence of internal damping mechanisms in the body is considered by decomposing the deformation gradient and the magnetic induction into ‘elastic’ and ‘viscous’ parts. Constitutive laws for material behaviour and evolution equations for the non-equilibrium fields are derived that agree with the laws of thermodynamics. To illustrate the theory the problems of stress relaxation, magnetic field relaxation, time dependent magnetic induction and strain are formulated and solved for a specific form of the constitutive law. The results, that show the effect of several modelling parameters on the deformation and magnetisation process, are illustrated graphically.     

Natural convection in unsteady MHD couette flow

 The combined effect of natural convection and uniform transverse magnetic field on the couette flow of an electrically conducting fluid between two parallel plates for impulsive motion of one of the plates in discussed. Under the assumption of negligible induced magnetic field and applied magnetic field being fixed relative to the fluid or plate, the governing equations have been solved exactly, and the expressions for velocity and temperature field have been presented for two different cases. A comparative study is made between the velocity field for magnetic field fixed with respect to plate and magnetic field fixed with respect to fluid. Received on 12 July 1999     

含孔软铁磁材料Mindlin板中弹性波散射问题

基于考虑磁弹相互作用的Mindlin板弯曲波动方程，采用波函数展开法，分析研究 了含孔软铁磁材料Mindlin板中弹性波散射与动应力集中问题，给出了问题的分析 解和数值算例. 通过分析发现：磁感应强度对动弯矩集中系数和动剪力集中系数有 增加的作用，特别是在低频的情况下.     

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     

带剪切型压电激励器的板的高阶剪切变形理论

利用压电材料固有的正，逆压电效应可以对结构变形和振动进行控制。与外加电场与极化方向平行于板厚度的压电材料的拉伸作动机制相比，外加电场与极化方向垂直的压电材料的剪切作动机制可以在作动器内产生较小的应力，从而降低作动器边界产生分层破坏的危险。本文对于压电材料的剪切作动机制进行研究，应用三阶剪切变形理论建立带剪切型压电激励器的智能层合板模型。采用哈密顿原理导出带剪切型压电激励器的层合板的控制方程。采用空间法得到了各种边界条件组合条件下板的解析解。数值算例对一三层板采用高阶和一阶剪切变形理论进行计算，结果表明两种理论所得的变形曲线很相似。但对于厚度剪切型激励器而言，由于激励器是引起板的剪切变形，而高阶剪切变形理论比一阶剪切变形理论能更好地反映结构的剪切应变能，因此高阶剪切变形理论可以提供板变形的更为精确的解。因此，对于厚度剪切型激励器，剪切变形理论的选取对于板变形结果的好坏有重要的作用。