Determining the Axisymmetric, Geometrically Nonlinear, Thermoelastoplastic State of Laminated Orthotropic Shells

A technique is developed to determine the axisymmetric, geometrically nonlinear, thermoplastic stress–strain state of laminated ortotropic shells of revolution under loads that cause a meridian stress state and torsion. The technique is based on the rectilinear-element hypotheses for the whole stack of layers. The active elastoplastic deformation of an ortotropic material is described by deformation-type equations that have been derived without resort to the existence conditions for the plastic potential. The scalar functions in the constitutive equations depend on the intensity of shear strains and temperature. The problem is solved through the numerical integration of a system of differential equations. The technique is tried out in designing tubular specimens subjected to axial force and torque. As an example, the elastoplastic state of a corrugated shell is analyzed     

Thermoelastoplastic State of Flexible Laminated Shells under Axisymmetric Loading Along Various Plane Paths

The paper presents a technique for thermoelastoplastic stress–strain analysis of flexible laminated shells of revolution under complex axisymmetric loading. The constitutive deformation equations are used to describe loading along arbitrary plane paths. The problem is solved by the method of successive approximations. A numerical example is given     

Thermoelastoplastic Stress—Strain State of Laminated Transversely Isotropic Shells under Axisymmetric Loading

A method is proposed for determining the thermoelastoplastic stress–strain state of laminated shells of revolution, made of isotropic and transversely isotropic materials, under axisymmetric loading. The method is based on the Kirchhoff–Love hypotheses for a layer stack, the theory of deformation along paths of small curvature for isotropic materials, and Hill's flow theory with isotropic hardening for transversely isotropic materials. The loading history is accounted for. The problem is solved by the method of successive approximations. Numerical examples are given     

Axisymmetric Thermoelastoplastic Stress State of Isotropic Solids of Revolution Under Impulsive Loading

Consideration is given to the solution of a dynamic problem for a solid of revolution with an arbitrary meridional section under impulsive thermomechanical loading inducing elastoplastic strains. The theory of small elastoplastic deformations is used. The constitutive equations are linearized by the variable-parameter method. The unloading process is described by a linear law. The solution technique involves the finite-element approximation in spatial coordinates and the finite-difference representation of time derivatives. Based on the principle of linear summation, recurrent relations are derived for successive evaluation of nodal displacements by an explicit scheme in time. Solution for cylinders and disks are presented to illustrate the influence of elastoplastic deformations on wave processes     

Superposition Method in Thermal-Stress Problems for Rectangular Plates

The classical two-dimensional biharmonic problem for a rectangular domain is considered. Some historical aspects of the problem are elucidated. The superposition method turns out to be efficient in solving thermoelastic-equilibrium problems in a rectangle. The relationship between the mathematical and engineering approaches to these problems is studied. A few typical examples are given __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 101–117, September 2005.     

Flexure of Clamped Rectangular Plates Resting on Winkler Foundations

ABSTRACT

The flexure under uniform load of clamped rectangular plates resting on elastic foundations is treated via eigenfunction expansions. The expansion coefficients are determined by expanding each of the nonorthogonal eigen-functions in terms of an orthogonal set of real functions. The numerical convergence of the associated eigenfunction expansions for some aspect ratios of the plate and some values of the foundation modulus are presented.     

Thermoelastoplastic deformation of a thick-walled cylinder with a radial crack

The thermoelastoplastic fracture mechanics problem of a thick-walled cylinder subjected to internal pressure and a nonuniform temperature field is solved by the method of elastic solutions combined with the finite-element method. The correctness of the solution is provided by using the Barenblatt crack model, in which the stress and strain fields are regular. The elastoplastic problem of a cracked cylinder subjected to internal pressure and a nonuniform temperature field are solved. The calculation results are compared with available data. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 173–183, May–June, 2008.     

四边自由的矩形厚板的自由振动

由于要满足自由边的条件,使得本文所研究的板的分析相当困难。本文采用叠加法完满的解决了这个不易克服的困难,并使得边界条件能满足到任意精度,所取的位移解的每一项都严格满足控制微分方程。     

功能梯度矩形厚板的三维热弹性分析

直接从三维热弹性力学基本方程出发，通过引入两个位移函数和两个应力函数，导出了一个二阶的齐次状态方程和一个四阶的非齐次状态方程。分析中采用了层合近似模型，即将板划分成厚度足够小的若干薄层，从面可将每一层内的材料常数近似为常数。给出了任意厚度的四边简支横观各向同性功能梯度矩形板的热弹性分析，特别当板较薄时，与薄板理论进行了数值对比，发现两者结果吻合很好。最后研究了材料梯度指标对热弹性场的影响，结果显示梯度指标对热应力和位移都有着显著的作用：在不同的区间，梯度指标对它们有不同的影响；并且在同一区间，梯度指标对两者的影响程度也有所不同。     

阶梯式矩形板的振动

用奇异函数建立阶梯式矩形板自由振动和强迫振动的微分方程并求得其通解，用Ｗ算子给出振型函数的表达式及常见支承条件下板的频率方程，本文解可用于多种边界条件的板。     

四边简支矩形中厚板的弯曲

本文采用Reissner中厚板理论求解了四边简支矩形中厚板的弯曲问题。文中首先对Reissner中厚板理论的控制方程进行了适当的变更,使之成为非耦联的二阶偏微分方程组,然后利用有限积分变换法求解所得新的控制方程,得到了四边简支矩形中厚板受均布载荷作用下的解析解。文中所述方法可用以求解具有其它边界条件和载荷的矩形中厚板的弯曲问题,同时还可移植应用于其它中厚板理论。     

Nonlinear Oscillations of Viscoelastic Rectangular Plates

Nonlinear oscillations of viscoelastic simply supported rectangular plates are studied by assuming the Voigt–Kelvin constitutive model. Using Hamilton's principle in conjunction with the kinematics associated with Kirchhoff's plate model, the governing equations of motion including the effect of damping are represented in terms of the transversal deflection and a stress function. Utilizing the Bubnov–Galerkin method, the nonlinear partial differential equations are reduced to an ordinary differential equation which is studied geometrically by approximate construction of the Poincaré maps. Explicit expressions are given for periodic solutions.     

Spline-Approximation Method Applied to Solve Natural-Vibration Problems for Rectangular Plates of Varying Thickness

The natural vibrations of anisotropic rectangular plates of varying thickness with complex boundary conditions are studied using the spline-collocation and discrete-orthogonalization methods. The basic principles of the approach are outlined. The natural vibrations of orthotropic plates with parabolically varying thickness are calculated. The results (natural frequencies and modes) obtained with different boundary conditions are analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 10, pp. 90–99, October 2005.     

Elastoplastic Axisymmetric Stress-Strain State of Laminated Shells Made of Isotropic and Transversely Isotropic Materials with Different Moduli

A method is developed for analysis of the elastoplastic stress-strain state of laminated shells of revolution under axisymmetric loading. The shells are made of isotropic and transversally isotropic materials with different moduli. The method is based on the Kirchhoff-Love hypotheses for the whole laminate, the theory of deformation along paths of small curvature (for isotropic materials), and the theory of elasticity with different tensile and compressive moduli (transversely isotropic materials). The problem is solved by the method of successive approximations. Numerical examples are given __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 8, pp. 88–96, August 2005.     

点支撑预应力中厚矩形板的弯曲

基于Reissner-Mindlin一阶剪切变形板理论，讨论在预加面内机械荷载或温度场作用下，点支撑中厚矩形板的弯曲问题。温度场假定沿板表面为均布，沿板厚方向为线性分布的。利用考虑剪切变形影响的Timoshenko梁函数，采用Rayleigh-Ritz法给出不同边界条件下点支撑中厚板在横向荷载作用下的挠度和弯矩分布。结果表明，均匀温度场与预加面内压力将使板的挠度和弯矩增加。支撑点位置的变化、边界约束条件和横向剪切变形效应都对板的内力大小和分布有显著影响。     

Thermoelastoplastic deformation of noncircular cylindrical shells

A method to determine the nonstationary temperature fields and the thermoelastoplastic stress-strain state of noncircular cylindrical shells is developed. It is assumed that the physical and mechanical properties are dependent on temperature. The heat-conduction problem is solved using an explicit difference scheme. The temperature variation throughout the thickness is described by a power polynomial. For the other two coordinates, finite differences are used. The thermoplastic problem is solved using the geometrically nonlinear theory of shells based on the Kirchhoff-Love hypotheses. The theory of simple processes with deformation history taken into account is used. Its equations are linearized by a modified method of elastic solutions. The governing system of partial differential equations is derived. Variables are separated in the case where the curvilinear edges are hinged. The partial case where the stress-strain state does not change along the generatrix is examined. The systems of ordinary differential equations obtained in all these cases are solved using Godunov's discrete orthogonalization. The temperature field in a shell with elliptical cross-section is studied. The stress-strain state found by numerical integration along the generatrix is compared with that obtained using trigonometric Fourier series. The effect of a Winkler foundation on the stress-strain state is analyzed Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 79–90, August 2008.     

面内预应力功能梯度材料矩形板的横向弯曲

功能梯度材料（FGM)是一类具有广阔应用前景的新型复合材料。本文考虑材料物性参数随坐标和温度变化的特性，研究横向荷载和面内预加荷载作用下FGM矩形板在各种边界条件下的弯曲问题。给出了基于一维微分求积格式的Galerkin技术的半解析方法，并以ZrO2/Ti-6Al-4V板为例考察了材料组分，温度相关性，面内预加荷载，边界约束条件等对FGM板弯曲行为的影响。结果表明，FGM板的弯曲变形介于各向同性陶瓷板与各向同性金属板之间，且随板抗弯刚度的增大而逐步减小，在高温下条件下必须考虑材料物性和温度的相关性。     

Karman型正交异性矩形薄板非线性弯曲的统一求解方法

本文对Karman型四边支承正交异性薄板在5种不同边界条件下的几何非线性弯曲进行了统一分析。所设的位移函数均为梁振动函数。它们精确地满足边界条件，利用Galerkin方法和位移函数的正交属性，转换控制方程为非线性代数方程。用“稳定化双共轭梯度法”求解稀疏矩阵线性方程组以及“可调节参数的修正迭代法”求解非线性代数方程组，最后给出了相应的数值结果。     

Complex Parametric Vibrations of Flexible Rectangular Plates

In this paper we consider parametric oscillations of flexible plates within the model of von Kármán equations. First we propose the general iterational method to find solutions to even more general problem governed by the von Kármán–Vlasov–Mushtari equations. In the language of physics the found solutions define stress–strain state of flexible shallow shell with a bounded convex space R ² and with sufficiently smooth boundary . The new variational formulation of the problem has been proposed and his validity and application has been discussed using precise mathematical treatment. Then, using the earlier introduced theoretical results, an effective algorithm has been applied to convert problem of finding solutions to hybrid type partial differential equations of von Kármán form to that of the ordinary differential (ODEs) and algebraic (AEs) equations. Mechanisms of transition to chaos of deterministic systems with infinite number of degrees of freedom are presented. Comparison of mechanisms of transition to chaos with known ones is performed. The following cases of longitudinal loads of different sign are investigated: parametric load acting along X direction only, and parametric load acting in both directions X and Y with the same amplitude and frequency.