Nonlinear stress and deformation analysis of thin current-carrying strip-shells

The problems of nonlinear deformation of a thin current-carrying shell under the coupled action of an unsteady electromagnetic field and a mechanical field are studied. The nonlinear magneto-elastic kinetic equations, the physical equations, the geometric equations, the electrodynamics equations, and the expressions of Lorentz force of a thin current-carrying shell under the action of a coupled field are given. Normal Cauchy form nonlinear differential equations, which include ten basic unknown functions in all, are obtained by the variable replacement method. Using the difference and quasilinearization methods, the nonlinear magneto-elastic equations are reduced to a sequence of quasilinear differential equations, which can be solved by the method of discrete orthogonalization. Numerical solutions for the stresses and deformations in the thin current-carrying strip-shell with two simply supported edges are obtained by considering a specific example. The results that the stresses and deformations in a thin current-carrying strip-shell with two simply supported edges change with variation of the electromagnetic parameters are discussed, through a special case. It is shown that the deformations of the shell can be controlled by changing the electromagnetic parameters Published in Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 130–144, September 2007.     

Determining the stress state of flexible orthotropic shells of revolution in magnetic field

A two-dimensional nonlinear magnetoelastic model of a current-carrying orthotropic shell of revolution is constructed taking into account finite orthotropic conductivity, permeability, and permittivity. It is assumed that the principal axes of orthotropy are aligned with the coordinate axes and that the orthotropic body is magnetically and electrically linear. The coupled nonlinear differential equations derived describe the stress-strain state of flexible current-carrying orthotropic shells of revolution that have an arbitrarily shaped meridian and orthotropic conductivity and are in nonstationary mechanical and electromagnetic fields. A method to solve this class of problems is proposed Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 64–76, August 2008.     

电磁场中载流薄板的弯曲行为分析

在建立载流薄板的非线性磁弹性运动方程、电动力学方程和Lorentz力表达式的基础上,通过变量代换,将描述薄板的磁弹性状态方程整理成含有10个基本函数的标准Cauchy型.采用差分法及准线性化方法,将标准Cauchy型非线性偏微分方程组,变换成为能够用离散正交法求解的准线性微分方程组.通过算例,得到了电磁场与机械载荷联合...     

导电圆柱薄壳的热磁弹性效应分析

研究了磁场环境中受机械载荷作用的导电圆柱薄壳的热磁弹性问题．首先,根据电动力学方程和广义Ohm定律,得到了导电薄壳电流密度的分布,考虑到Joule热效应及热平衡方程,得到了导电薄壳的温度分布．其次,通过几何方程、物理方程、运动方程和电动力学方程导出了导电薄壳在机械场、电磁场以及温度场作用下的基本方程．最后,采用差分法及准线性化方法,得到了可以应用离散正交法求解的准线性微分方程组．对于导电圆柱薄壳,得到了Lorentz力表达式,并且推导了温度场积分特征值．讨论了导电圆柱薄壳应力、温度及变形随外加电磁参量的变化规律,并通过实例证实了可以通过改变电、磁、力场的参数来实现对薄壳的应力、应变、温度的控制．     

变厚度载流旋转壳体的热磁弹性分析

针对磁场环境中受机械载荷作用变厚度载流旋转壳体热磁弹性问题，基于薄壳的几何方程、物理方程、运动方程、电动力学方程，建立其热磁弹性基本方程．应用线性化方法，得到了关于热磁弹性问题的线性迭代方程，转化为包括8个基本未知量的标准型方程组．关于载流旋转壳体，给出Lorentz力表达式，导出了温度场及温度场积分特征值．分析了变厚度锥形旋转壳体应力、温度及变形随外加电磁参量的变化规律，并通过数值模拟对理论分析进行了验证．研究结果可为载流旋转壳体热磁弹性问题研究提供理论参考．     

有限厚度、大压入的球压试验材料力学响应

针对磁场环境中受机械载荷作用变厚度载流旋转壳体热磁弹性问题,基于薄壳的几何方程、物理方程、运动方程、电动力学方程,建立其热磁弹性基本方程．应用线性化方法,得到了关于热磁弹性问题的线性迭代方程,转化为包括8个基本未知量的标准型方程组．关于载流旋转壳体,给出Lorentz力表达式,导出了温度场及温度场积分特征值．分析了变厚度锥形旋转壳体应力、温度及变形随外加电磁参量的变化规律,并通过数值模拟对理论分析进行了验证．研究结果可为载流旋转壳体热磁弹性问题研究提供理论参考．     

Nonlinear problems of current-carrying round plates with varying thickness in the electromagnetic field

I.IntroductionInrecentdecades,thescholarsinphysicalmechanicscircleshavegivenmuchmoreattentiontothequestionsofthestressesanddeformationsoftheelasticbodiesunderthecoupledinfluenceofthemechanicalfieldandelectromagneticfield,whichresultsinthefoeinganddevelopm…     

基于热-磁-弹性耦合的二维导电薄柱壳数值分析

基于几何方程、物理方程、运动方程和电动力学方程,建立了二维导电薄柱壳热磁弹性基本方程．考虑到Joule热效应,引入热平衡方程及广义Ohm定律,得出导电薄柱壳的温度场．利用变量代换方法,整理成具有10个基本未知量的标准型方程组．采用差分法和准线性化方法,给出准线性微分方程组．对于二维导电薄柱壳,导出了Lorentz力表达式、温度场积分特征值．通过实例计算,得到了二维导电薄柱壳应力、位移、温度随外加电磁参量的变化规律．研究结果可为二维导电薄壳热磁弹性问题研究提供理论参考．     

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     

轴向运动导电薄板磁弹性耦合动力学理论模型

针对磁场环境中轴向运动导电薄板的动力学理论建模问题进行研究,得到较为完备的磁弹性耦合振动基本方程及相应的补充关系式。在考虑几何非线性效应下,给出薄板运动的动能、应变能以及外力虚功的表达式。应用哈密顿变分原理,推得磁场中轴向运动薄板的非线性磁弹性耦合振动方程,并得到力和位移满足的边界条件。基于麦克斯威尔电磁场方程,并考虑相应的电磁本构关系和电磁边界条件,推得任意磁场环境中轴向运动导电薄板满足的电动力学方程和所受电磁力表达式。分别针对纵向磁场环境、横向磁场环境、条形板等具体情形,给出了振动方程、电动力学方程和电磁力的简化形式。所得结果,可为此类问题的进一步求解和分析提供理论参考。     

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.     

内共振条件下轴向运动梁磁弹性超谐共振研究

以载流导线激发的磁场中轴向运动梁为研究对象,同时考虑外激励力作用,推导出梁的磁弹性非线性振动方程．通过位移函数的设定和伽辽金积分法,将非线性振动方程离散为常微分方程组．采用多尺度法得到系统的近似解析解．应用Matlab 和Mathematica 软件求解幅频响应方程,并对稳态解进行稳定性判定．通过具体算例得到前两阶假设模态的响应幅值随不同参数的变化规律．结果发现：系统在内共振条件下发生超谐波共振时,二阶假设模态幅值明显小于一阶;随着外激励的增大,多值解区域范围明显缩小;随着电流强度增加,振动幅值减小,表明载流导线能够起到控制共振的作用．     

Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

In this article, the nonlinear dynamic responses of sandwich functionally graded(FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell's nonlinear shell theory and Hamilton's principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters,specifically, the radial load, core thickness, foam type, foam coefficient, structure damping,and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.     

Magnetoelastic bending and stability of current-carrdying coil structures

As a results of magnetoelastic interaction, the mechanical behavior of current-carrying coil structures, such as deformation and instability, is a key problem in the design of strong field magnets. In this paper, a nonlinear mathematical model is presented to described the deformation and buckling of D-type current-carrying coil, based on the Biot-Savart law and the bending theory of curved beams. The bending deformation, the critical value of current for the magnetoelastic buckling of the current-carrying coil, and the effect of the type and number of supports at middle part of the bendling coil on the critical value are quantitatively investigated by a semi-analytical and semi-numerical method. The numerical results are shown to be in good agreement with the experimental data. The project supported by the National Natural Science Foundation of China, the Science Foundation of the State Education Committee of China for Outstanding Teacher in Universities, and the Natural Science Foundation of Gansu Province of China     

磁场中旋转运动圆板的分叉与混沌研究

应用数值模拟方法研究磁场中旋转运动圆板的分叉与混沌问题。首先,基于薄板理论和麦克斯韦电磁场方程组,给出了动能、应变势能、外力虚功以及电磁力的表达式,再利用哈密顿原理,得到磁场中旋转运动圆板横向振动的非轴对称非线性磁弹性振动微分方程组。其次,采用贝塞尔函数作为圆板的振型函数进行伽辽金积分,得到了轴对称情况下横向振动的常微分方程组表达式。最后,针对主共振,取周边夹支边界条件的圆板作为算例,得到了当振型函数取一阶时,将磁感应强度、外激励振幅和激励频率作为控制参数的分叉图及庞加莱映射图等计算结果,并讨论了分叉参数对系统的分叉与混沌的影响。数值计算结果表明,这些控制参数的变化影响系统稳定性,在分叉参数逐渐变化的过程中,系统经历从混沌到多倍周期运动再到混沌的往复过程。     

Determining linear vibration frequencies of a ferromagnetic shell

The problems of determining the roots of dispersion equations for free bending vibrations of thin magnetoelastic plates and shells are of both theoretical and practical interest, in particular, in studying vibrations of metallic structures used in controlled thermonuclear reactors. These problems were solved on the basis of the Kirchhoff hypothesis in [1–5]. In [6], an exact spatial approach to determining the vibration frequencies of thin plates was suggested, and it was shown that it completely agrees with the solution obtained according to the Kirchhoff hypothesis. In [7–9], this exact approach was used to solve the problem on vibrations of thin magnetoelastic plates, and it was shown by cumbersome calculations that the solutions obtained according to the exact theory and the Kirchhoff hypothesis differ substantially except in a single case. In [10], the equations of the dynamic theory of elasticity in the axisymmetric problem are given. In [11], the equations for the vibration frequencies of thin ferromagnetic plates with arbitrary conductivity were obtained in the exact statement. In [12], the Kirchhoff hypothesis was used to obtain dispersion relations for a magnetoelastic thin shell. In [5, 13–16], the relations for the Maxwell tensor and the ponderomotive force for magnetics were presented. In [17], the dispersion relations for thin ferromagnetic plates in the transverse field in the spatial statement were studied analytically and numerically.In the present paper, on the basis of the exact approach, we study free bending vibrations of a thin ferromagnetic cylindrical shell. We obtain the exact dispersion equation in the form of a sixth-order determinant, which can be solved numerically in the case of a magnetoelastic thin shell. The numerical results are presented in tables and compared with the results obtained by the Kirchhoff hypothesis. We show a large number of differences in the results, even for the least frequency.     

Buckling of thick cylindrical shells under external pressure: A new analytical expression for the critical load and comparison with elasticity solutions

In this paper a set of stability equations for thick cylindrical shells is derived and solved analytically. The set is obtained by integration of the differential stability equations across the thickness of the shell. The effects of transverse shear and the non-linear variation of the stresses and displacements are accounted for with the aid of the higher order shell theory proposed by [Voyiadjis, G.Z. and Shi, G., 1991, A refined two-dimensional theory for thick cylindrical shells, International Journal of Solids and Structures, 27(3), 261–282.]. For a thick shell under external hydrostatic pressure, the stability equations are solved analytically and yield an improved expression for the buckling load. Reference solutions are also obtained by solving numerically the differential stability equations. Both the full set that contains strains and rotations as well as the simplified set that contains rotations only were solved numerically. The relative magnitude of shear strain and rotation was examined and the effect of thickness was quantified. Differences between the benchmark solutions and the analytic expressions based on the refined theory and the classical shell theory are analysed and discussed. It is shown that the new analytic expression provides significantly improved predictions compared to the formula based on thin shell theory.     

Incremental elastic motions superimposed on a finite deformation in the presence of an electromagnetic field

The equations describing the interaction of an electromagnetic sensitive elastic solid with electric and magnetic fields under finite deformations are summarized, both for time-independent deformations and, in the non-relativistic approximation, time-dependent motions. The equations are given in both Eulerian and Lagrangian form, and the latter are then used to derive the equations governing incremental motions and electromagnetic fields superimposed on a configuration with a known static finite deformation and time-independent electromagnetic field. As a first application the equations are specialized to the quasimagnetostatic approximation and in this context the general equations governing time-harmonic plane-wave disturbances of an initial static configuration are derived. For a prototype model of an incompressible isotropic magnetoelastic solid a specific formula for the acoustic shear wave speed is obtained, which allows results for different relative orientations of the underlying magnetic field and the direction of wave propagation to be compared. The general equations are then used to examine two-dimensional motions, and further expressions for the wave speed are obtained for a general incompressible isotropic magnetoelastic solid.     

Stress Analysis of Anisotropic Inhomogeneous Shells of Revolution Subjected to Centrifugal Forces

An approach is proposed to analyze the stress state of thin shells of revolution under centrifugal loads with regard for anisotropy, the meridional variability of geometrical and mechanical parameters, and the eccentricity of the axis of revolution relative to the axis of geometric symmetry. Allowance is also made for the change in the dimensions of the shell due to deformation, which results in a nonaxisymmetric distribution of stresses and strains and their nonlinear dependence on the squared frequency of rotation. By separating variables, the problem is made one-dimensional and then solved numerically. The stress state of an ellipsoidal filament-winded composite shell is analyzed     

双轴对称截面薄壁圆弧曲梁的弹性稳定平衡方程

基于薄壁构件分析的基本假定,采用双轴对称截面薄壁圆弧曲梁的精确翘曲位移表达式,导出了曲梁考虑几何非线性情况下的总势能,根据欧拉公式得到了曲梁的稳定平衡方程。推导中采用横截面线性和非线性总应变为零的假定,从而无需考虑横向应力的影响,对应变高阶项采用合理的简化处理,使理论推导过程简单明了。在理论推导的基础上分析了简支拱在均布径向荷载和两端等弯矩荷载作用下的平面内和平面外屈曲问题,并与其他研究者的结果进行了比较,追溯了各理论结果存在差别的根源,论证了本文理论推导过程的合理性。使用通用有限元软件ANSYS进行了模拟,与本文的分析结果一致,证明了所得公式的正确性。通过一些无碍结果的近似使所得公式形式简洁,便于在工程中应用。