有初始几何缺陷的一般壳体的非线性应变分量公式

本文从非线性三维连续介质的应变分量公式出发,导出具有初始几何缺陷一般薄壳的非线性应变分量公式,在推导过程中没有局限于任何一种特定的壳体,因此公式具有一般性.这组公式可以为研究有初始几何缺陷的壳体几何非线性问题提供应变几何学理论基础.     

磁场作用功能梯度壳体磁弹耦合动力学模型

针对电磁场环境中金属-陶瓷功能梯度圆柱壳体结构，基于物理中面下的几何关系和Hooke定律，确定了圆柱薄壳体的非线性本构关系.根据Kirchhoff-Love弹性理论，给出了非均质弹性壳体的变形应变能、动能及其变分运算式.基于电磁弹性理论，得出了电磁场作用下磁性功能梯度壳体所受涡流Lorentz力和磁化力模型.应用Hamilton广义变分原理，建立功能梯度薄壳体的磁弹性耦合非线性振动方程组，得出了描述功能梯度结构的具有变形场与电磁场耦合特征的动力学理论模型.通过对磁场中功能梯度壳体固有振动问题的举例分析，得到了壳体振动特征方程和固有频率变化规律，表明磁场和材料体积分数指数的增大能够使频率值减小，而在周向波数影响曲线中出现频率最小值的情形.研究方法可为多场耦合系统理论建模及动力学分析提供参考.     

黏弹性圆柱壳计及切变形和转动惯量时的动力学稳定性

根据修正的Timoshenko理论,在几何非线性中考虑了剪切变形和转动惯量,对黏弹性圆柱壳的动力稳定性进行了研究.根据Bubnov-Galerkin法,结合基于求积公式的数值方法,将问题简化为求解具有松弛奇异核的非线性积分-微分方程的问题.针对物理-力学和几何参数在大范围内的变化,研究壳体的动力特性,显示了材料的黏弹性对圆柱壳动力稳定性的影响.最后,比较了通过不同的理论得到的结果.     

旋转壳在子午面内整体弯曲的几何非线性摄动有限元法及在波纹管计算中的应用(Ⅰ)

为切实有效地计算波纹管,建立了旋转壳在子午面内整体弯曲几何非线性问题的摄动有限元法。以结构环向应变的均方根为摄动小参数,将有限元节点位移列式和节点力列式直接展开。通过摄动小参数将非线性有限元的载荷分级和迭代过程有机地统一起来,即载荷的分级是有约束的,每一级载荷增量和所对应的位移增量之间的关系是已知的,每一级的计算一步到位。为叙述方便并具实用性,将旋转壳用截锥壳单元进行离散。位移分量和载荷分量沿环向按Fourier级数展开,沿子午线用多项式插值,端面弯矩和横向力化成载荷分量离散到节点上。采用Sanders中小转角非线性几何方程和各向同性广义Hooke定律。对多层材料叠合而成的旋转壳按各层薄膜应变、弯曲应变、扭转应变相等的原则进行处理,该方法能方便有效地计算单层和多层波纹管整体纯弯曲、横向弯曲的几何非线性问题。并为有限元处理非线性问题提供了一条新途径。     

弹塑性状态下Ⅰ型裂纹前缘应力应变场的奇异性分析

本文从弹塑性力学的三维基本方程出发,分析了幂硬化材料Ⅰ型裂纹前缘应力、应变场的奇异性,发现,裂尖附近诸应力、应变分量的奇异性沿厚度不变;六个应力分量的奇异性不完全相同,六个应变分量的奇异性也不完全相同.     

薄壳非线性变形理论

对薄壳的非线性变形,给出了应变与位移之间的精确关系.经过合理的简化,给出了壳的挠度与厚度同级的大变形基本公式.当薄壳为无限长柱形且作柱形变形时,精确地求得了壳的挠度与长度同级的大变形基本公式.     

随动强化材料壳体的安定分析

在循环加载下壳体结构的安定分析,特别是对于具有应变强化的材料制成的壳体结构的安定分析具有很大的实际意义.文中对随动强化材料的安定定理有了进一步的认识并应用它去分析壳体结构的安定载荷.对于一个真实状态其残余应力与塑性应变之间是相关的.但我们在定理中所示的与时间无关的残余应力场(σ_ijr)和与时间无关的几何容许的塑性应变场(σ_ijp)可以是不相关的.明确指出这点对于工程应用带来很多方便,否则将是十分困难的.为此还给出了该定理的新的证明方法.我们还应用了上述定理对一个半球封头的圆柱壳体进行了安定分析.根据所求得的弹性解,各种可能的残余应力和塑性应变分布,结构的安定分析可归结为一个数学规划问题.计算结果表明应变强化材料的安定载荷要比理想塑性材料的安定载荷高出30～40%,这说明在安定分析中考虑材料强化是重要的,可使壳体结构的设计承载能力有相当大的提高,同时对改进目前壳体结构的设计提供了科学依据.     

定常围压作用时平面应变理想塑性体Mises屈伏条件的精确解

把应力函数引入平面问题的Mises屈状条件后那个二阶非线性偏微分方程分解为两个二阶线性偏微分方程,用柯西积分公式求出这两个方程右端的已知函数,然后解这个方程,由此定出弹塑性区域的分界线和求出塑性区内的各应力分量,给出一个例题说明本文方法的应用.     

旋转壳在子午面内整体弯曲的几何非线性摄动有限元肖及在波纹管计算中的应用（Ⅰ）

为切实有效地计算波纹管，建立了旋转壳在子午面内整体弯曲几何非线性问题的摄动有限元法。以结构环向应变的均方根为摄动小参数，将有限元节点位移列式和节点力列式直接展开。通过摄动小参数将非线性有限元的载荷分级和迭代过程有机地统一起来，即载荷的分级是有约束的，每一级载荷增量和所对应的位移增量之间的关系是已知的，每一级的计算一步到位。为叙述方便并具实用性，将旋转壳用截锥壳单元进行离散。位移分量和载荷分量沿环向按Fourier级数展开，沿子午线用多项式插值，端面弯矩和横向力化成载荷分量离散到节点上。采用Sanders中小转角非线性几何方程和各向同性广义Hooke定律，对多层材料叠合而成的旋转壳按各层薄膜应变、弯曲应变、扭转应变相等的原则进行处理，该方法能方便有效地计算单层和多层波纹管整体纯弯曲、横向弯曲的几何非线性问题。并为有限元处理非线性问题提供了一条新途径。     

一种新的基于非线性相位的Fourier理论及其应用

信号的正频率表示自Fourier分析诞生以来一直都是物理学家、数学家以及信号分析工作者密切关注的问题.基于调和分析和复分析方法,在过去近二十年里诞生了单分量函数理论以及基于单分量函数的函数(信号)表示理论.作为原创性理论这个方法将信号快速分解为一些具有正的非线性瞬时频率的基本信号之和.该理论植根于经典数学并可以推广到定义在高维流形上的向量值及矩阵值信号.这从而也创立了高维空间中的有理逼近理论.单分量函数理论包括正瞬时频率的数学定义及几个最重要的单分量函数类的刻画.单分量函数的表示理论包括核心自适应Fourier分解(Core Adaptive Fourier Decomposition,或Core AFD)及其若干变种,包括解绕AFD,循环AFD,再生核Hilbert空间的预一正交AFD.除了理论及方法的概述,本文也给出了两个新证明:迄今最一般的依据极大选择原理的自适应分解的收敛性的证明;以及参数重复选择的及用到再生核导数的必要性的证明.最后我们给出该理论与数学及信号分析中若干相关理论的联系,以及该方法的某些应用.     

Problem-Oriented Functionals in the Theory of Nonlinearly Elastic Composite Shells

Based on the Kirchhoff-Love or Timoshenko hypotheses and with regard for a possible membrane or shear degeneration, mixed linearized functionals for four variants of shell theory are presented. The convergence of numerical methods is improved by choosing small strain components as additional variable functions. New classes of problems for thin and nonthin shells are solved. The stress-strain state of shells is studied using different variants of this theory.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

Geometrically nonlinear shell finite element based on the geometry of a planar curve

An engineering approach for constructing a curved triangular finite element of a thin shell is considered. The approach is based on the assumption that the triangle sides are planar nearly circular curves before and after deformation. A geometrically nonlinear formulation of a triangular finite element of a thin Kirchhoff–Love shell is given. The predictive capabilities of the element are tested using benchmark problems of nonlinear deformation of elastic plates and shells.     

A variational principle in a geometrically nonlinear theory of heterogeneous viscoelastic shells

The use of the hereditary theory for shells heterogeneous across their thickness is considered. A variational method is formulated for calculating thin anisotropic shells made of a material whose deformation behavior can be described by relations of the linear theory of viscoelasticity. In order to transform the corresponding functional into a form suitable for shells, some assumptions related to concepts of the theory of thin shells are introduced. In the capacity of Euler equations, physical relations, nonlinear equilibrium equations, and nonlinear boundary conditions are derived. The state equations are deduced for a multilayered shell. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 231–240, March–April, 2009.     

A nonlinear shell model of Koiter's type

We define a new two-dimensional nonlinear shell model “of Koiter's type” that can be used for the modeling of any type of shell and boundary conditions and for which we establish an existence theorem. The model uses a specific three-dimensional stored energy function of Ogden's type that satisfies all the assumptions of John Ball's fundamental existence theorem in three-dimensional nonlinear elasticity and that is adapted here to the modeling of thin nonlinearly elastic shells by means of specific deformations that are quadratic with respect to the transverse variable.     

A finite-variation method in nonlinear shell mechanics

A numerical algorithm is proposed for calculating coefficients of first-and second-order variations of strain energy in a nonlinear finite-element model of a shell, which are necessary to define equilibrium states of the shell and investigate stability of the states. Several numerical schemes are considered based on various finite-difference approximations. For these schemes, the accuracy, convergence, and computation time are analyzed using popular geometrically nonlinear problems of deformation of elastic plates and shells.     

Axisymmetric deformation of bimetallic shells of revolution in the supercritical region

Geometrically nonlinear relationships of the theory of thin layered shells are applied to analyze axisymmetric strain of bimetallic shells of revolution in a temperature field. One-dimensional nonlinear boundary-value problems are solved by a combination of the linearization method and the discrete orthogonalization method. A numerical approach is proposed to solve the boundary-value problems in the supercritical strain region.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 72, pp. 52–56, 1990.     

变厚度扁锥壳的非线性固有频率

借助于变厚度圆薄板非线性动力学变分方程和协调方程,给出了变厚度扁薄锥壳的非线性动力学变分方程和协调方程·　假设薄膜张力由两项组成,将协调方程化为两个独立的方程,选取变厚度扁锥壳中心最大振幅为摄动参数,采用摄动变分法,将变分方程和微分方程线性化·　对周边固定的圆底变厚度扁锥壳的非线性固有频率进行了求解;一次近似得到了变厚度扁锥壳的线性固有频率,三次近似得到了变厚度扁锥壳的非线性固有频率,且绘出了固有频率与静载荷、最大振幅、变厚度参数的特征曲线图·　为动力工程提供了有价值的参考·     

A Geometric Theory of Nonlinear Morphoelastic Shells

Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms, are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.