基于Nitsche方法与拟牛顿求解的二维接触问题等几何分析

在等几何框架内,基于Nitsche方法推导了二维无摩擦弹性接触列式,采用基于BFGS逆更新的拟牛顿迭代格式求解.提出了Nitsche接触列式中罚系数的经验公式和拟牛顿求解时迭代的初始化方法,研究了基于割线刚度阵的修正方法以克服因接触面变化而导致的迭代发散.所提出的接触分析方法在粗糙网格下也能精确描述接触边界,列式推导简单,计算量小.算例表明了接触列式和求解方法的有效性.     

张力索杆结构最小势能分析方法与结构特性研究

张力索杆结构为柔性多态体系,计算方法复杂。本文基于势能最小化迭代建立平衡态的最小势能方法,推导了体系势能、下降向量及步长列式,建立势能最小化共轭梯度法迭代格式,并用VC++编程实现了算法。通过正交四边形网格索网数值分析算例验证了程序正确性,并分析了新型六边形网格马鞍形双曲索网和索杆张拉整体结构的非线性荷载特性。研究表明最小势能迭代分析是一种有效的柔性张力结构非线性分析方法。     

具有大范围回转和伸缩运动的柔性机械臂动力学建模与分析

大范围转动和伸缩运动是某些机械臂系统的典型运动特征,如快堆中燃料组件转运机构.为了实现精准定位操作,设计过程中需要考虑机械臂的弹性振动问题．本文从刚体运动和弹性变形耦合的角度出发,运用广义Hamilton原理建立了一种变长度回转梁的时变动力学方程．通过变量替换,利用假设模态法得到二阶变系数微分方程,并采用Newmark-β法进行求解．根据计算结果,讨论了不同回转角速度和轴向伸缩情形对机械臂末端弹性振动的影响,并说明了在此类柔性机械臂振动分析中考虑刚弹耦合的必要性．     

几何可变体系大变形机构模态的二元体构建法

针对仅有一个自由度的几何可变体系,基于大变形机构模态的特性,提出常变、瞬变的判断方法,进而对常变体系提出大变形机构模态的二元体构建算法。该算法将问题简化为类似于一元二次方程的求解,无须求解非线性超越方程组,可给出完备的模态解答和完整的解答路径及其分叉点。本法具有原理直观、公式简洁、解答完备的优点,可作为结构力学几何构造分析教学的补充内容。     

基于运动约束解过约束并联机构变形协调方程

提出利用运动约束关系来间接求解过约束并联机构变形协调方程．首先介绍了该方法的原理，接着分别针对平面和空间过约束并联机构，详述该方法的解决步骤，结果验证了该方法的正确性，从中还可看出该方法在求解复杂过约束并联机构时非常简洁，最后介绍了采用该方法解决多度过约束问题．     

CONSTRUCTION OF MECHANISM MODES OF GEOMETRICALLY UNSTABLE SYSTEMS IN LARGE DEFORMATION

针对仅有一个自由度的几何可变体系,基于大变形机构模态的特性,提出常变、瞬变的判断方法,进而对常变体系提出大变形机构模态的二元体构建算法。该算法将问题简化为类似于一元二次方程的求解,无须求解非线性超越方程组,可给出完备的模态解答和完整的解答路径及其分叉点。本法具有原理直观、公式简洁、解答完备的优点,可作为结构力学几何构造分析教学的补充内容。     

基于动力松弛法的松弛索杆体系找形分析

索杆张力结构施工成形分析需要解决一个松弛态索杆系统的平衡形态求解问题,该问题可归结为一个给定构件原长的受荷索杆机构系统的找形问题。文中利用动力松弛法进行该类松弛索杆体系的找形分析。由于不建立刚度矩阵,避免了体系几何不稳定性所引起的刚度矩阵奇异问题。该方法采用悬链线索单元模型,可以考虑索大垂度效应。文中还推导了反映索原长和内力之间关系的悬链线索单元协调方程。通过一个正交松弛索网算例分析,考察了该找形方法的计算精度和收敛性。最后还模拟了一个索穹顶的施工成形过程,表明了该找形方法用于索杆张力结构施工成形分析的有效性。     

基于能量优化的桁架几何非线性问题力密度法

力密度法最初是求解膜结构找形问题的方法,经发展可用于计算桁架结构的几何非线性问题。本文应用力密度法建立结构变形后的非线性平衡方程及相应的雅可比矩阵,用于迭代求解;从能量原理出发,推导出杆单元应变能、外荷载势能、结构总势能在每次迭代位移方向上关于步长λ的显式列式。相对于固定步长的牛顿法,本文将最优迭代步长λ引入求解,使结构在每次迭代位移方向上均达到总势能最小。经桁架算例验证,表明该方法可加快计算收敛进程。     

一种求解粘弹性大变形问题的时域分段展开算法

摘要： 利用展开技术,提出一种求解粘弹性大变形问题的时域分段自适应算法,建立了显式递推格式的Update-Lagrange有限元求解模式,只需在初始时刻迭代求解一个瞬时弹性大变形问题,其它时段均不需迭代计算,并可通过展开阶数对计算精度的控制,实现时域的自适应计算,数值算例表明所提算法可行有效。     

非均匀弹性材料中反平面的运动裂纹

用解析方法研究了非均匀弹性材料中反平面运动裂纹问题。首先采用余弦变换求解非均匀材料的基本方程，然后根据混合边值条件建立裂纹运动的对偶积分方程，再把对偶积分方程化为第二类Fredholm积分方程。给出了数值算例，计算结果表明材料的非均匀性对动应力强度因子有较大的影响。     

Stabilization of beam parametric vibrations with shear deformations and rotary inertia effects

The purpose of this theoretical work is to present a stabilization problem of beam with shear deformations and rotary inertia effects. A velocity feedback and particular polarization profiles of piezoelectric sensors and actuators are introduced. The structure is described by partial differential equations with time-dependent coefficient including transverse and rotary inertia terms, general deformation state with interlaminar shear strains. The first order deformation theory is utilized to investigate beam vibrations. The beam motion is described by the transverse displacement and the slope. The almost sure stochastic stability criteria of the beam equilibrium are derived using the Liapunov direct method. If the axial force is described by the stationary and continuous with probability one process the classic differentiation rule can be applied to calculate the time-derivative of functional. The particular problem of beam stabilization due to the Gaussian and harmonic forces is analyzed in details. The influence of the shear deformations, rotary inertia effects and the gain factors on dynamic stability regions is shown.     

Stability conditions for straightened, non-linearly elastic, annular cylindrical sectors

The plane-strain straightening deformation of certain elastic annular cylindrical sectors is considered in conjunction with the closely related deformation of uniaxial extension of elastic rectangular strips. For the case of mixed boundary conditions of place and traction, sufficient and necessary conditions for the infinitesimal stability of these deformations are provided and several illustrative examples are analysed.     

From Non-Linear Elasticity to Linearized Theory: Examples Defying Intuition

Material frame indifference implies that the solution in non-linear elasticity theory for a connected body rigidly rotated at its border is a rigid, stress-free, deformation. If the same problem is considered within linear elasticity theory, considered as an approximation to the true elastic situation, one should expect that if the angle of rotation is small, the body still undergoes a rigid deformation while the corresponding stress, though not zero, remains consistently small. Here, we show that this is true, in general, only for homogeneous bodies. Counterexamples of inhomogeneous bodies are presented for which, whatever small the angle of rotation is, the linear elastic solution is by no means a rigid rotation (in a particular case it is an “explosion”) while the stress may even become infinite. If the same examples are re-interpreted as problems in an elasticity theory based upon genuinely linear constitutive relations which retain their validity also for finite deformations, it is shown that they would deliver constraint reaction forces that are not in equilibrium in the actual, deformed, state. This furnishes another characterization of the impossibility of an exact linear constitutive theory for elastic solids with zero residual stress.      

On two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic bimodulus materials

The paper compares two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic materials with different elastic moduli in tension and compression. The approaches use different nonlinear constitutive equations describing the elastic deformation of the transversally isotropic bimodulus materials. Both approaches employ the theory of deformation along paths of small curvature and the Kirchhoff-Love hypothesis for the whole laminate to describe the deformation of the isotropic materials. The problem is solved by the method of successive approximations. The solutions of specific problems obtained by the two approaches are analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 59–69, June 2008.     

耦合变形对大范围运动柔性梁动力学建模的影响

柔性梁在作大范围空间运动时,产生弯曲和扭转变形,这些变形的相互耦合形成了梁在纵向以及横向位移的二次耦合变量。本文考虑了变形产生的几何非线性效应对运动柔性梁的影响,在其三个方向的变形中均考虑了二次耦合变量,利用弹性旋转矩阵建立了准确的几何非线性变形方程,通过Lagrange方程导出系统的动力学方程。仿真结果表明,在大范围运动情况下,仅在纵向变形中计及了变形二次耦合量的一次动力学模型,与考虑了完全几何非线性变形的模型具有一定的差异。     

一类刚-梁耦合系统平面稳态运动的稳定性

研究了中心力场中的一类刚-弹耦合系统的平面运动动力学，模型是带有一悬臂 梁的刚体. 综合考虑了系统轨道运动与姿态运动，在Lagrange力学体系下给出了系统的运 动方程，在保守系统和考虑梁的材料黏滞阻尼两种情况下，利用能量-动量方法给出了一类 相对平衡点稳定性的充分条件.     

Mechanics and thermodynamics of fluid-saturated highly elastic materials

We consider a mixture that consists of a highly elastic material and a liquid dissolved in this material. Using the laws of classical thermodynamics, we state a variational principle describing the mixture equilibrium under static loading conditions. From this principle, we derive equilibrium equations and a system of constitutive relations characterizing the mixture elastic and thermodynamic properties. We state problems describing the stress-strain state of a swollen material and a statically loaded material in thermodynamic equilibrium with the liquid. We consider the case of incompressible mixture. The general theory is illustrated by examples concerned with the deformation behavior of inhomogeneously swollen cross-linked polymers and with their thermodynamics of strains and swelling in solvent media.     

The Method of R-Functions in the Solution of Elastic Problems on the Basis of Reissner's Mixed Variational Principle

A method is presented for solving boundary-value elastic problems on the basis of the variational–structural method of R-functions and Reissner's mixed variational principle. A mathematical formulation is given to problems on the deformation of elastic bodies under mixed boundary conditions and bodies interacting with smooth rigid dies. Solutions satisfying all the boundary conditions are proposed. For undetermined components of these solutions, the resolving equations are derived and their properties are studied. A posteriori estimation of numerical solutions is made. As examples, solutions are found to a problem on the stress–strain state of a short cylinder and to a contact problem on a cylinder interacting with a smooth die. A numerical method of solving such problems is analyzed for convergence, and the accuracy of the solutions is estimated.     

Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations

The dynamic analysis of a generalized linear elastic body undergoing large rigid rotations is investigated. The generalized linear elastic body is described in kine- matics through translational and rotational deformations, and a modified constitutive relation for the rotational deformation is proposed between the couple stress and the curvature tensor. Thus, the balance equations of momentum and moment are used for the motion equations of the body. The floating frame of reference formulation is applied to the elastic body that conducts rotations about a fixed axis. The motion-deformation coupled model is developed in which three types of inertia forces along with their incre- ments are elucidated. The finite element governing equations for the dynamic analysis of the elastic body under large rotations are subsequently formulated with the aid of the constrained variational principle. A penalty parameter is introduced, and the rotational angles at element nodes are treated as independent variables to meet the requirement of C1 continuity. The elastic body is discretized through the isoparametric element with 8 nodes and 48 degrees-of-freedom. As an example with an application of the motion- deformation coupled model, the dynamic analysis on a rotating cantilever with two spatial layouts relative to the rotational axis is numerically implemented. Dynamic frequencies of the rotating cantilever are presented at prescribed constant spin velocities. The maximal rigid rotational velocity is extended for ensuring the applicability of the linear model. A complete set of dynamical response of the rotating cantilever in the case of spin-up maneuver is examined, it is shown that, under the ultimate rigid rotational velocities less than the maximal rigid rotational velocity, the stress strength may exceed the material strength tolerance even though the displacement and rotational angle responses are both convergent. The influence of the cantilever layouts on their responses and the multiple displacement trajectories observed in the floating frame is simultaneously investigated. The motion-deformation coupled model is surely expected to be applicable for a broad range of practical applications.     

作大范围运动弹性梁刚—柔耦合动力学建模

利用弹性梁的变形理论和 Hamilton力学原理对作大范围运动弹性梁的刚 -柔耦合动力学建模理论进行了研究。分析了大范围运动对弹性梁的横向振动和纵向振动的影响 ,得到了大范围运动与弹性梁的中线耦合变形之间的耦合作用对该系统动力学性质有显著的影响 ,从而提出了作大范围运动弹性梁的刚柔耦合动力学模型