功率型变分原理和功能型拟变分原理及其应用

自从钱伟长建立了功率型变分原理以来,功率型变分原理和功能型变分原理在理论方面和应用方面有什么区别和联系,成为学术界关注的课题.应用变积方法,根据Jourdain原理和d’Alembert原理,建立了不可压缩黏性流体力学的功率型变分原理和功能型拟变分原理,推导了不可压缩黏性流体力学的功率型变分原理的驻值条件和功能型拟变分原理的拟驻值条件.研究了不可压缩黏性流体力学的功率型变分原理在有限元素法中的应用.研究表明,功率型变分原理与Jourdain原理相吻合,功能型变分原理与d’Alembert原理相吻合.功率型变分原理直接在状态空间中研究问题,不仅在建立变分原理的过程中可以省略在时域空间中的一些变换,而且给动力学问题有限元素法的数值建模带来方便.     

有限变形的极限分析定理

本文建立了有限变形的极限分析的变分原理(定理1),证明了它与有限变形的极限分析的全套方程和条件等价.本文又证明了:根据此变分原理求得的有限变形的极限载荷乘子,介于有限变形的极限分析的上限定理和下限定理所分别给出的上限解和下限解之间.     

大变形对称弹性理论的广义变分原理

本文以陈至达提出的变形几何非线性理论￣［１］为基础，应用Ｌａｇｒａｎｇｅ乘子法，对大变形对称弹性力学问题进行了研究，给出了相应的位能广义变分原理、余能广义变分原理，以及动力学问题的广义变分原理；同时，文中还证明了位能广义变分原理和余能广义变分原理的等价性。     

用变分方法求解大变形对称弹性力学问题

文中以经典力学的数学理论和陈氏定理为基础,用变分的方法求解大变形对称弹性力学问题,得出了以瞬时位形为基准的位能广义变分原理和余能广义变分原理,以及两个变分原理的等价性;此外,还给出了以瞬时位形为基准的动力学问题的广义变分原理.     

钱氏定理在有限变形极矩弹性力学广义变分原理的应用

应用Lagrange乘子法和钱伟长证明的两类广义变分原理的等价定理,在本文中导出有限变形极矩弹性力学的广义变分原理.文中采用了在拖带坐标系描述法建立的有限变形应变张量(称为Biot有限变形应变定义的准确形式)和应变速率定义与拖带系应力张量构成完整的数学描述.     

超弹性矩形板单向拉伸时微孔的增长*

本文研究了含中心微孔的超弹性矩形板在单向拉伸时的有限变形和受力分析.为了考察微孔的存在对矩形板变形和应力的影响,将问题化成一个超弹性环形板的变形和受力分析,并用最小势能原理得到变分近似解.进行了数值计算,分析了微孔的增长情况.     

有限变形弹性理论随机变量变分原理及有限元法

本文将材料、载荷、结构几何形状、力和位移边界条件的随机性,直接引入有限变形弹性理论的泛函变分表达式中,应用小参数摄动法,建立了统一的随机变量变分原理及非线性随机有限元法,并将其应用于结构可靠性分析。算例表明,应用此方法处理随机变量的力学问题,具有使程序实施简便,计算效率高等优点。     

大应变固结理论的分区变分原理及其广义变分原理

土体材料本构特性的差异问题与大变形问题是分析岩土材料变形特性的基本问题．根据有限变形的描述方法构筑土体结构大变形固结方程,证明了大变形固结的变分原理A·D2应用分区子结构的连续条件,推导固结理论的分区变分原理．引用Lagrange乘子法构筑并证明了大变形固结问题在无约束状态下的广义分区变分原理．     

复合材料加筋薄壁圆锥壳体有限变形的混合型理论

本文利用变分原理和平均筋条刚度法,建立了在任意载荷作用下纵向和环向密加筋复合材料圆锥壳体有限变形的Donnell型理论.考虑了面板最一般的弯曲拉伸耦合关系和加筋筋条的偏心效应的影响.导出了平衡条件、边界条件和变形协调方程.给出了以应力函数和挠度函数表示的耦合形式的非性性变系数偏微分方程组.对于一些特殊情况,给出了相应的简化方程.     

大变形弹塑性率变分极值原理

在Updated Lagrangian率形式下,研究了大变形弹塑性率问题的对偶极值变分原理.证明了变分泛函的凸性取决于一个所谓的间隙函数.     

A variational arbitrary Lagrangian-Eulerian finite element formulation for standard dissipative solids at finite strains

A novel Arbitrary Lagrangian-Eulerian (ALE) finite element formulation for standard dissipative media at finite strains is presented. In contrast to previously published ALE approaches accounting for dissipative phenomena, the proposed scheme is fully variational. Consequently, no error estimates are necessary and thus, linearity of the problem and the corresponding Hilbert-space are not required. Hence, the resulting Variational Arbitrary Lagrangian-Eulerian (VALE) finite element method can be applied to highly nonlinear phenomena as well. In case of standard dissipative solids, so-called variational constitutive updates provide a variational principle. Based on these updates, the deformation mapping follows from minimizing an incrementally defined (pseudo) potential, i.e., energy minimization is the overriding criterion that governs every aspect of the system. Therefore, it is natural to allow the variational principle to drive mesh adaption as well. Thus, in the present paper, the discretizations of the deformed as well as the undeformed configuration are optimized jointly by minimizing the respective incremental energy of the considered mechanical system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

大变形非线性弹性力学的广义变分原理

本文导出了大变形非线性弹性力学的两个具有σ_ij,e_ij,和u_i三类独立变量的广义变分原理,证明了当应力应变关系为约束条件时这两个广义变分原理是等价的.文中对某些特例也作了阐明.     

Towards variational constitutive updates for finite strain plasticity models based on non-associative evolution equations

In this contribution, first steps towards variational constitutive updates for finite strain plasticity theory based on non-associative evolution equations are presented. These schemes allow to compute the unknown state variables such as the plastic part of the deformation gradient, together with the deformation mapping, by means of a fully variational minimization principle. Therefore, standard optimization algorithms can be applied to the numerical implementation leading to a very robust and efficient numerical implementation. Particularly, for highly non-linear, singular or nearly ill-posed physical models like that corresponding to crystal plasticity showing a large number of possible active slip planes, this is a significant advantage compared to standard constitutive updates such as the by now classical return-mapping algorithm. While variational constitutive updates have been successfully derived for associative plasticity models, their extension to more complex constitutive laws, particularly to those featuring non-associative evolution equations, is highly challenging. In the present contribution, a certain class of non-associative finite strain plasticity models is discussed and recast into a variationally consistent format. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A fully variational algorithmic formulation for wrinkling at finite strains

This contribution is concerned with an efficient novel algorithmic formulation for wrinkling at finite strains. In contrast to previously published numerical implementations, the advocated method is fully variational. More precisely, the parameters describing wrinkles or slacks, together with the unknown deformation mapping, are computed jointly by minimizing the potential energy of the considered mechanical system. Furthermore, the wrinkling criteria are naturally included within the presented variational framework. The presented approach allows to employ three–dimensional constitutive models directly, i.e., plane stress conditions characterizing membranes are variationally enforced by minimizing the potential energy with respect to the transversal strains. Since the proposed formulation for wrinkling in membranes is fully variational, it can be conveniently combined with other variational methods (based on energy minimization). As an example, a variationally consistent framework for finite strain plasticity theory is considered. More precisely, the minimization principle characterizing wrinkling in elastic membranes and that describing plasticity in inelastic solids are coupled leading to a novel variational approach for inelastic membranes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

有限变形弹性圆杆中的孤波

在同时引入横向惯性和横向剪切应变的情况下,导出了有限变形弹性圆杆的非线性纵向波动方程,方程中包含了二次和三次的非线性项以及由横向剪切与横向惯性导致的两种几何弥散效应．借助Mathematica软件,利用双曲正割函数的有限展开法,对该方程和对应的截断的非线性方程进行求解,得到了非线性波动方程的孤波解,同时给出了这些解存在的必要条件．