A general expression of magnetic force for soft ferromagnetic plates in complex magnetic fields

The experiments of ferromagnetic plates in different magnetic environments exhibit two distinct phenomena, i.e. the magnetoelastic instability of a ferromagnetic plate in transverse magnetic fields, and the increase of natural frequency of a ferromagnetic plate with low susceptibility in an inplane magnetic field. Although these two typical phenomena can be predicted separately by two kinds of theoretical models in which the magnetic forces are formulated by totally different expressions, no theoretical model has been found to commonly describe them. This makes it difficult to predict theoretically magnetoelastic interaction of a ferromagnetic structure in complex magnetic environment. A variational principle, here, is proposed to establish the governing equations of magnetoelastic interaction for soft ferromagnetic thin plate structures under complex magnetic fields. The functional is chosen as the summation of the magnetic energy and the strain energy as well as the external work from applied magnetic fields. From manipulations of the variational principle, the governing equations of the magnetic field and mechanical deformation together with an expression of equivalent magnetic force exerted on the ferromagnetic plates are obtained. It is shown that this theoretical model can commonly characterize the experimental phenomena of the magnetoelastic interaction aforementioned.     

Magneto-thermo-elasticity of an electroconductive circular cylindrical shell featuring nonlinear deformations

The magneto-thermo-elastic interactions of a circular cylindrical shell immersed in applied magnetic and thermal fields are modeled and the shell’ nonlinear response and stability are investigated. The shell is made of a transversely isotropic, perfectly electroconductive non-ferromagnetic material. The shell’s deformation is characterized by small strains and moderately small rotations. A geometrically nonlinear, first-order transverse shear deformation shell model is developed and Galerkin’s method is used for spatial semi-discretization. The close connection between the static nonlinear response and the temperature-frequency interaction is explored to interpret the multi-solutions of the nonlinear response. Influence of prebuckling deformations on the buckling results is addressed and the significance of the applied magnetic fields on the response and stability is evaluated.     

A differential quadrature analysis of dynamic and quasi-static magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field

A rapid, convergent and accurate differential quadrature method (DQM) is employed for numerical simulation of dynamic and quasi-static magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field. Fundamental equations of plane electromagnetic, temperature and elastic fields are formulated. To the best of the authors’ knowledge, this is the first attempt to apply the DQM to magneto-thermo-elasticity and the first attempt to analyze a finite two-dimensional magneto-thermo-elastic problem. The fundamental equations and the inhomogeneous time-dependent boundary conditions are discretized in spatial and temporal domain by differential quadrature (DQ) rules. The unknowns satisfying the governing equations, the boundary and initial conditions simultaneously are computed in the entire domain by means of DQM with high efficiency and accuracy using dramatically less grid points in both spatial and time domain. Solutions of magnetic field, eddy current, temperature change and dynamic solutions of stresses and deformations are illustrated graphically.     

Dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting infinite plate subjected to an arbitrary variation of magnetic field

Summary In the present paper, dynamic and quasi-static behaviors of magneto-thermo-elastic stresses in a conducting infinite plate subjected to an arbitrary variation of the magnetic field are investigated. It is assumed that a magnetic field defined by an arbitrary function of time acts on both side surfaces of the infinite plate in the direction parallel to its surfaces. Fundamental equations of one-dimensional electromagnetic, temperature and elastic fields are formulated. Then, solutions of magnetic field, eddy current, temperature change and both dynamic solutions and quasi-static ones of stresses and deformations in the infinite plate are derived analytically. The solutions of stresses are determined to be sums of thermal stress caused by eddy current loss and magnetic stress caused by Lorentz force. For the case that the arbitrary function is given by the sine function, the dynamic and quasi-static behaviors of the stresses are examined by numerical calculations.     

夹芯梁结构有效性能的多尺度变分渐近模型

为有效分析夹芯梁结构性能,基于变分渐近法建立多尺度变分渐近模型。首先基于旋转张量分解概念建立三维夹芯梁几何非线性的能量方程;利用梁结构细长和非均质较小的特征,将三维夹芯梁结构各向异性、非均质问题严格分解为宏观层面的梁轴线的一维非线性分析和细观层面的单胞本构分析。基于最小势能原理,通过对单胞应变能泛函变分主导项最小化得到有效属性和波动函数解,代入梁的一维模型进行全局非线性响应分析。利用得到的全局响应、波动函数解重构局部场。由于变分特性,构建多尺度模型可以很容易通过有限元数值实现。通过三类夹芯梁结构算例表明:构建模型得到的全局位移和局部应力场与三维有限元具有很好的一致性,但计算成本和建模工作量明显减少,为结构设计人员在初始设计阶段对夹芯梁结构性能评估提供了一种简洁的途径。      

Magneto-thermo-elastic response in a perfectly conducting medium with three-phase-lag effect

This paper deals with the problem of magneto-thermo-elastic interactions in an unbounded, perfectly conducting elastic medium due to the presence of periodically varying heat sources in the context of linear theory of generalized thermo-elasticity with energy dissipation (TEWED or GN-III model), without energy dissipation (TEWOED or GN-II model) and three-phase-lag model (3P model). The governing equations of generalized thermo-elasticity of the above models under the influence of a magnetic field are established. The Laplace-Fourier double transform technique has been used to get the solution. The inversion of the Fourier transform has been done by using residual calculus, where poles of the integrand are obtained numerically in a complex domain by using Laguerre’s method, and the inversion of the Laplace transformation is done numerically using a method based on Fourier series expansion technique. Displacement, temperature, stress and strain distributions have been computed numerically and presented graphically in numbers of figures. A comparison of the results for different theories (GN-II, GN-III and 3P model) and the effect of magnetic field and damping coefficient on the physical quantities has been discussed.     

复合材料热传导性能的变分渐近均匀化细观力学模型

为准确预测非均质复合材料的有效热导率和局部温度场分布,采用单胞变分渐近均匀化方法构建了一种新的细观力学模型。首先从非均质连续体热传导变分问题入手,使用变分渐近法将其细观力学模型转换为约束条件下泛函的最小化——取驻值问题;使用有限元法(FEM)推导了离散形式能量泛函的最小化求解过程;根据宏观性能(如全局温度及相应的梯度和波动函数)重构单胞的局部温度场和热通量。采用多个二元复合材料算例验证了所构建理论和程序的有效性和准确性。     

Phase field simulations of low-dimensional ferroelectrics

A coupled theory of nonlinear electro-thermo-viscoelasticity with inclusion of hysteresis, aging and damage effects is developed based on non-equilibrium thermodynamics. In consideration of the Gibbs free energy including the contribution of the free electric field as a functional of the histories of stress, temperature, temperature gradient and electric field in the reference configuration with damage being introduced as an internal state variable, constitutive relations and kinetic laws are obtained from the energy balance equation and the entropy production inequality. Finite electro-thermo-viscoelasticity and nonlinear electro-thermo- elasticity can be taken as special cases. A superposition principle of time, aging, temperature, stress and electric field is proposed for materials with memory on an intrinsic time scale so that the long-term property functions may be represented with horizontal and vertical shifting of the momentary master curves. This formulation lays a foundation for the characterization of the short- and long-term behavior of time-dependent materials with evolving damage under combined electric, thermal and mechanical loadings.     

Nonlinear zigzag theory for electrothermomechanical buckling of piezoelectric composite and sandwich plates

Summary A coupled geometrically nonlinear efficient zigzag theory is presented for electrothermomechanical analysis of hybrid piezoelectric plates. The geometric nonlinearity is included in Von Karman sense. The thermal and potential fields are approximated as piecewise linear across sublayers. The deflection accounts for the transverse normal strain due to thermal and electric fields. The inplane displacements are considered to have layerwise variations, but are expressed in terms of only five primary displacement variables, independent of the number of layers. The coupled nonlinear equations of equilibrium and the boundary conditions are derived from a variational principle. The nonlinear theory is used to obtain the initial buckling response of symmetrically laminated hybrid plates under inplane electrothermomechanical loading. Analytical solutions for buckling of simply-supported plates under thermoelectric load are obtained for comparing the results with the available exact three-dimensional (3D) piezothermoelasticity solution. The comparison establishes that the present results are in excellent agreement with the 3D solution, when the pre-buckling transverse normal strain is neglected in the latter solution. The present results are also compared with the third order theory with the same number of displacement variables to highlight the positive effects of the layerwise terms in the displacement field approximations of the zigzag theory.     

功能梯度扁球壳在热-机械荷载作用下的屈曲分析

基于经典壳理论,应用虚功原理和变分法推导了均匀变温场中的功能梯度圆底扁球壳在均布外侧压力作用下的位移型几何非线性控制方程。考虑固定夹紧边界条件,运用打靶法计算获得了球壳轴对称变形的数值结果。考察了材料体积分数指数、组分材料弹性模量和均匀变温场对壳体平衡路径,上、下临界荷载以及平衡构型的影响。数值结果表明,随材料体积分数指数的增加和组分材料弹性模量的减小,壳体上、下临界荷载均会显著减小。均匀升温,会使壳体上临界荷载显著增加,下临界荷载轻微减小。为方便工程设计人员进行几何、材料、荷载和变温参数的选取,给出了一个实用数表和一些实用的数值曲线。     

A variational approach to the problem of solidification of a metal semi-infinite thermally nonlinear body

Summary Based upon the Gauss differential variational principle and the method of optimal linearization, analytical approximate solutions are developed for the problem of solidification of a metal semi-infinite body whose coefficient of thermal conductivity depends linearly on the temperature. For the case of constant thermal properties, the approximate solution obtained by means of the Gauss variational principle agrees remarkable well with the exact solution obtained by Neumann.With 1 Figure     

Reflection of magneto-thermo-elastic waves from a rotating elastic half-space

Models of generalized magneto-thermo-elasticity based on the Lord-Shulman theory (LS) with one relaxation time, the Green-Lindsay theory (GL) with two relaxation times, as well as the classical dynamic coupled theory (CD), are used to study the electro-magneto-thermo-elastic interactions in a semi-infinite perfectly conducting solid. The elastic medium rotates with uniform angular velocity. The initial magnetic field is parallel to the boundary of the half-space. Reflection of magneto-thermo-elastic waves under generalized thermo-elasticity theory is studied. The reflection coefficients are obtained.     

A variational approach to a circular hyperelastic membrane problem

Summary The variational principles of nonlinear elasticity are applied to a problem of axially symmetric deformation of a uniform circular hyperelastic membrane. The supported edge of the membrane is in a horizontal plane and its radius is equal to that of the undeformed plane reference configuration, so that an initially plane unstretched membrane is subjected to a dead load due to its weight.It is shown how the stationary complementary energy principle can be used to obtain an accurate approximate solution for the deformation and stress distribution. It is also shown how the potential energy principle can be applied to the problem and how close bounds for an energy functional can be obtained from the two theorems. Numerical results are presented for realistic properties for a rubberlike material and for two strain energy functions, the semi-linear and the neo-Hookean.     

On the effectiveness of variation in the physical variables on the steady MHD flow between parallel plates with heat transfer

The influence of variation in physical variables on the steady Hartmann flow with heat transfer is studied. An external uniform magnetic field is applied perpendicular to the parallel plates and the fluid is acted upon by a constant pressure gradient. The viscosity and the thermal and electric conductivities are assumed to be temperature dependent. The two plates are kept at two constant but different temperatures and the viscous and Joule dissipations are considered in the energy equation. A numerical solution for the governing non‐linear coupled equations of motion and the energy equation is obtained. The effect of magnetic field, the temperature dependent viscosity, thermal conductivity, and electric conductivity on both the velocity and temperature distributions is examined. Copyright © 2005 John Wiley & Sons, Ltd.     

The Variational Form of the Mathematical Model of a Thermal Explosion in a Solid Body with Temperature-Dependent Thermal Conductivity

The variational form of a mathematical model of a thermal explosion has been developed based on a variational formulation of a nonlinear problem of stationary thermal conductivity in a homogeneous solid body. The model takes the temperature dependence of the thermal conductivity of a solid body into account. The presented example of quantitative analysis of the model demonstrates a method for finding the combination of parameters for determining a thermal explosion in a plate with an exponential temperature dependence of the thermal conductivity. At the same time, the analysis allows one to identify the number of steadystate temperature distributions inside a body whose energy release intensifies with a temperature increase.     

金属芯压电纤维/聚合物复合材料压电-黏弹-塑性行为的细观力学模型

为了表征金属芯压电纤维增强聚合物基（MPF/PM）复合材料非线性、时变的压电-黏弹-塑性行为,基于变分渐近理论建立MPF/PM增量形式的细观力学模型。首先分别导出聚合物和MPF增量型本构方程,基于汉密尔顿扩展原理推导出MPF/PM压电-黏弹-塑性变分原理的能量泛函。考虑材料的时变和非线性特征,建立与求解瞬时有效机-电耦合矩阵有关的增量过程,并通过有限元技术实现模型的数值模拟。利用构建模型研究了不同铝芯体积分数、电场变化率和加载条件对MPF/PM有效全局应力-应变和单轴纵向拉伸性能的影响。结果表明,构建的模型能准确模拟MPF/PM多场耦合作用下的非线性、时变行为,为该新型智能材料的实际工程应用奠定理论基础。     

Magneto-thermo-elastic problem of a conducting rectangular cylinder subjected to sinusoidal change in time of magnetic field

This study is concerned with a magneto-thermo-elastic problem of a conducting rectangular cylinder in an external magnetic field which varies sinusoidally in time. Analytical solutions of eddy currents induced in the cylinder are derived on the basis of a theory of the quasi-stationary current. Two-dimensional transient temperature change produced by the eddy current loss is derived by means of the Green’s function method. The stresses in the infinitely long cylinder in a plane state are derived by making use of the Airy’s stress function. The effect of a frequency in a sinusoidal change in time of the external magnetic field on behaviors of eddy current loss, temperature change and stresses in the cylinder is examined by numerical calculation. The skin effect with an increase in a frequency of the external magnetic field on the transient response of temperature change and stresses in the cylinder is discussed.     

脉冲磁场激励下铁磁梁式板动力响应特征研究

基于广义变分原理得到磁力计算模型和磁场摄动技术,研究了不可移简支铁磁梁式板在矩形和锯齿形两类典型脉冲磁场作用下的磁弹性动力响应特征及动力失稳现象。数值模拟了铁磁梁式板在稳恒磁场,脉冲磁场下的振幅,频率的变化及其动力失稳特征。揭示了脉冲磁场的形状,脉冲持续时间,脉冲幅值与铁磁梁式板振幅,频率,临界失稳磁场值等特征参量之间的关系,为工程实际中铁磁结构的安全设计和可靠性评估提供依据和参考。     

Transient magnetic field and temperature modeling in large magnetapplications

A coupled magnetic/thermal model for studying heat and magnetic field diffusion in conducting materials subject to time-varying external fields has been developed, using the General Dynamics Thermal Analyzer code. Applications include energy storage devices, pulsed power transformers, and electromagnetic launchers, and the time scales of interest range from a magnetic field pulse of a microsecond to hundreds of seconds. The results of this model, showing the time and spatial variation of the magnetic field and temperature, are discussed for the projectile of an electromagnetic launcher