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

The paper analyzes the nonlinear deformation of a current-carrying thin shell in coupled electromagnetic and mechanical fields. The nonlinear magnetoelastic kinetic equations, physical equations, geometric equations, electrodynamic equations, expressions for the Lorentz force of a current-carrying thin shell in a coupled field are given. The normal Cauchy form nonlinear differential equations that include ten basic unknown functions are obtained by the variable replacement method. The difference and quasi-linearization methods are used to reduce the nonlinear magnetoelastic equations to a sequence of quasilinear differential equations that can be solved by discrete orthogonalization. Numerical solutions for the stresses and strains in a current-carrying thin strip shell with two edges simply supported are obtained as an example. The dependence of the stresses and strains in the current-carrying thin strip shell on the electromagnetic parameters is discussed. In a special case, it is shown that the deformation of the shell can be controlled by changing the electromagnetic parameters     

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．     

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．     

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.     

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.     

An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells

We analyze the steady-state response of a functionally graded thick cylindrical shell subjected to thermal and mechanical loads. The functionally graded shell is simply supported at the edges and it is assumed to have an arbitrary variation of material properties in the radial direction. The three-dimensional steady-state heat conduction and thermoelasticity equations, simplified to the case of generalized plane strain deformations in the axial direction, are solved analytically. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are then solved by the power series method. In the present formulation, the cylindrical shell is assumed to be made of an orthotropic material, although the analytical solution is also valid for isotropic materials. Results are presented for two-constituent isotropic and fiber-reinforced functionally graded shells that have a smooth variation of material volume fractions, and/or in-plane fiber orientations, through the radial direction. The cylindrical shells are also analyzed using the Flügge and the Donnell shell theories. Displacements and stresses from the shell theories are compared with the three-dimensional exact solution to delineate the effects of transverse shear deformation, shell thickness and angular span.     

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。     

ANALYSES ON NONLINEAR COUPLING OF MAGNETO-THERMO-ELASTICITY OF FERROMAGNETIC THIN SHELL-Ⅱ: FINITE ELEMENT MODELING AND APPLICATION

Based on the generalized variational principle of magneto-thermo-elasticity of a ferromagnetic thin shell established (see, Analyses on nonlinear coupling of magneto-thermo-elasticity of ferromagnetic thin shell-Ⅰ), the present paper developed a finite element modeling for the mechanical-magneto-thermal multi-field coupling of a ferromagnetic thin shell. The numerical modeling composes of finite element equations for three sub-systems of magnetic, thermal and deformation fields, as well as iterative methods for nonlinearities of the geometrical large-deflection and the multi-field coupling of the ferromagnetic shell. As examples, the numerical simulations on magneto-elastic behaviors of a ferromagnetic cylindrical shell in an applied magnetic field, and magneto-thermo-elastic behaviors of the shell in applied magnetic and thermal fields are carried out. The results are in good agreement with the experimental ones.     

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.     

四边简支功能梯度圆锥壳的非线性振动分析

研究了四边简支条件下功能梯度圆锥壳的非线性自由振动。首先,通过Voigt模型和幂律分布模型描述了功能梯度材料的物理属性。然后,考虑von-Karman几何非线性建立了功能梯度圆锥壳的能量表达式,利用Hamilton原理推出圆锥壳的运动方程。在此基础上,采用Galerkin法,只考虑横向振动,功能梯度圆锥壳运动方程可简化为单自由度非线性振动微分方程。最后,通过改进的L-P法和Runge-Kutta法求解非线性振动方程,讨论功能梯度圆锥壳的非线性振动响应,分析几何参数和陶瓷体积分数指数对圆锥壳非线性频率响应的影响。结果表明,几何参数对非线性频率和响应的影响相较于陶瓷体积分数指数更明显;圆锥壳的几何参数和陶瓷体积分数指数通过改变非线性频率影响振动响应;功能梯度圆锥壳呈弹簧渐硬非线性振动特性。     

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．     

载流圆锥薄壳的磁弹性应力与变形分析

在建立旋转壳体的非线性磁弹性运动方程的基础上,研究了电磁场和机械载荷联合作用下载流圆锥薄壳的磁弹性效应．通过算例,得到了载流圆锥薄壳的位移及应力与通电电流强度之间的关系．解决了圆锥薄壳顶点处的奇异性问题,给出了轴对称条件下的数值解．计算结果表明：改变通电电流强度,可以改变载流圆锥薄壳的应力与变形状态,达到控制圆锥薄壳的受力与变形的目的．     

考虑几何与压电非线性行为压电层合轴对称圆板的精确解

研究可移简支及夹支边界条件下,轴对称压电层合圆板在强电场和机械荷载联合作用下的非线性变形.考虑电致伸缩的非线性压电效应及几何非线性的影响,导出轴对称压电层合圆板的控制方程.通过调整坐标轴的位置对控制方程进行简化,得到关于挠度和径向力的4阶非线性控制方程.再通过简单的积分并引入无量刚变量将控制方程等价地化为2阶非线性耦合微分方程组.利用幂级数法得到可移简支及夹支边界条件下强电场和均布荷载共同作用时的挠度、径向力及径向位移的幂级数精确解.通过对双、单压电晶片执行器的数值计算及分析,得到电场、外载对于位移、径向力的影响关系.     

Transverse vibration characteristics of axially moving viscoelastic plate

The dynamic characteristics and stability of axially moving viscoelastic rect- angular thin plate are investigated.Based on the two dimensional viscoelastic differential constitutive relation,the differential equations of motion of the axially moving viscoelastic plate are established.Dimensionless complex frequencies of an axially moving viscoelastic plate with four edges simply supported,two opposite edges simply supported and other two edges clamped are calculated by the differential quadrature method.The effects of the aspect ratio,moving speed and dimensionless delay time of the material on the trans- verse vibration and stability of the axially moving viscoelastic plate are analyzed.     

Elasticity solution for free vibration analysis of four-parameter functionally graded fiber orientation cylindrical panels using differential quadrature method

In this paper, three-dimensional free vibrations analysis of a four-parameter functionally graded fiber orientation cylindrical panel is presented. The panel is simply supported at the edges and assumed to have an arbitrary variation of fiber orientation in the radial direction. A generalization of the power-law distribution presented in literature is proposed. Symmetric and asymmetric fiber orientation profiles are studied in this paper. Suitable displacement functions that identically satisfy the boundary conditions at the simply supported edges are used to reduce the equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which can be solved by differential quadrature method to obtain the natural frequency. The main contribution of this work is to illustrate the influence of the power-law exponent, of the power-law distribution choice and of the choice of the four parameters on the natural frequencies of continuous grading fiber orientation cylindrical panels. Numerical results are presented for a cylindrical panel with arbitrary variation of fiber orientation in the shell’s thickness and compared with discrete laminates composite panels. It is shown maximum natural frequencies will be obtained by using symmetric fiber orientation profiles.     

Large flexural vibrations of thermally stressed layered shallow shells

A dynamic nonlinear theory for layered shallow shells is derived by means of the von Karman-Tsien theory, modified by the generalized Berger-approximation. Moderately thick shells with polygonal planform composed of multiple perfectly bonded layers are considered. The shell edges are assumed to be prevented from in-plane motions and are simply supported. A distributed lateral force loading is applied to the structure, and additionally, the influence of a static thermal prestress, corresponding to a spatial distribution of cross-sectional mean temperature, is taken into account. In the special case of laminated shells made of transversely isotropic layers with physical properties symmetrically distributed about the middle surface, a correspondence to moderately thick homogeneous shells is found. Application of a multi-mode expansion in the Galerkin procedure to the governing differential equation, where the eigenfunctions of the corresponding linear plate problem are used as space variables, renders a coupled set of ordinary time differential equations for the generalized coordinates with cubic as well as quadratic nonlinearities. The nonlinear steady-state response of shallow shells subjected to a time-harmonic lateral excitation is investigated and the phenomenon of primary resonance is studied by means of the perturbation method of multiple scales. A unifying non-dimensional representation of the nonlinear frequency response function is presented that is independent of the special shell planform.     

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

I.IntroductionInrecentdecades,thescholarsinphysicalmechanicscircleshavegivenmuchmoreattentiontothequestionsofthestressesanddeformationsoftheelasticbodiesunderthecoupledinfluenceofthemechanicalfieldandelectromagneticfield,whichresultsinthefoeinganddevelopm…     

Magneto-elastic combination resonances analysis of current-conducting thin plate

Based on the Maxwell equations, the nonlinear magneto-elastic vibration equations of a thin plate and the electrodynamic equations and expressions of electro- magnetic forces are derived. In addition, the magneto-elastic combination resonances and stabilities of the thin beam-plate subjected to mechanical loadings in a constant transverse magnetic filed are studied. Using the Galerkin method, the corresponding nonlinear vibration differential equations are derived. The amplitude frequency response equation of the system in steady motion is obtained with the multiple scales method. The excitation condition of combination resonances is analyzed. Based on the Lyapunov stability theory, stabilities of steady solutions are analyzed, and critical conditions of stability are also obtained. By numerical calculation, curves of resonance-amplitudes changes with detuning parameters, excitation amplitudes and magnetic intensity in the first and the second order modality are obtained. Time history response plots, phase charts, the Poincare mapping charts and spectrum plots of vibrations are obtained. The effect of electro-magnetic and mechanical parameters for the stabilities of solutions and the bifurcation are further analyzed. Some complex dynamic performances such as period- doubling motion and quasi-period motion are discussed.     

Analysis and calculation of the nonlinear stability of the rotational composite shell

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