参变量变分原理的提出、发展与应用

参变量变分原理及其参数二次规划算法是由钟万勰院士1985年针对弹性接触边界非线性问题首次提出来的，经过将近40年的不断发展，目前参变量变分原理已经成功应用于各个领域，其中包括弹塑性分析、接触问题、润滑力学、岩土力学、变刚度杆系结构、先进材料性能分析、材料的蠕变与损伤、柔性结构力学和LQ最优控制等各个工程领域。本文首先回顾了参变量变分原理的起源，介绍了参变量变分原理的基本概念，然后以弹塑性分析问题为例，阐明建立参变量变分原理的理论模型以及实现数值参数二次规划求解原理，最后详细回顾了参变量变分原理的基本理论与相应数值算法在各个领域的发展及其工程应用，展示了参变量变分原理在求解各类非线性问题的特色与优势。     

分析力学初值问题的一种变分原理形式

明确了分析力学初值问题的控制方程，按照广义力和广义位移之间的对应关系，将 各控制方程卷乘上相应的虚量，代数相加，进而在 原空间中建立了分析力学初值问题的一种变分原理形式，即建立了分析力学初值问题的卷积 型变分原理和卷积型广义变分原理. 推导了分析力学初值问题卷积型变分原理和卷积型广义 变分原理的驻值条件. 在建立分析力学初值问题的一种变分原理形式的同时, 将变积方法推广为卷变积方法.     

The parametric variational principle for elastoplasticity

A parametric variational principle, the parametric minimum potential energy principle (abbreviated to PMPEP), is presented for the elastoplastic problems. The principle proposed is free from the restraint of Drucker's postulate and consequently suitable for solving the nonassociated plastic flow problems in rock, soil, concrete and other geomaterials.     

Generalized variational principles of the viscoelastic body with voids and their applications

From the Boltzmann‘ s constitutive law of viscoelastic materials and the linear theory of elastic materials with voids, a constitutive model of generalized force fields for viscoelastic solids with voids was given. By using the variational integral method, the convolution-type functional was given and the corresponding generalized variational principles and potential energy principle of viscoelastic solids with voids were presented. It can be shown that the variational principles correspond to the differential equations and theinitial and boundary conditions of viscoelastic body with voids. As an application, a generalized variational principle of viscoelastic Timoshenko beams with damage was obtained which corresponds to the differential equations of generalized motion and the initial and boundary conditions of beams. The variational principles provide a way for solving problems of viscoelastic solids with voids.     

广义参变本构模型分析

本文考察了力和(或)位移受给定条件控制的结构单元。这种控制条件可由线性化折线型本构关系表征,其本构关系是广义的,包括弹性、塑性、接触及其组合单元。借助于参变量变分原理和分级方法,具有广义本构关系单元的结构可以得解。所提出的方法还可直接用于解断裂问题。     

岩土中弹塑性渗透固结问题的参变量变分原理

本文给出渗透固结过程的参变量变分原理,可以用来处理岩土弹塑性渗透固结问题,并不受 Drucker 假设的限制,对于关连或非关连塑性流动情形皆可。     

Involutory transformations and variational principles with multi-varoables in thin plate bending problems

In this paper, the generalizd variational principles of plate bending, froblems are established from their minimum potential energy principle and minimum complementary energy principle through the elimination of their constraints by means of the method of Lagrange multipliers. The involutory transformations are also introduced in order to reduce the order of differentiations for the variables in the variation. Funhermore, these involutory transformations become infacl the additional constraints in the varialion. and additional Lagrange multipliers may be used in order to remove these additional constraints. Thus, various multi-variable variational principles are obtained for the plate bending problems. However, it is observed that. nol all the constrainls ofva’iaticn can be removed simply by the ordinary method of linear Lagrange multipliers. In such cases, the method of high-order Lagrange multipliers are usedto remove iliose constrainls left over by ordinary linear multiplier method. And consequently. some funct ionals of more general forms are oblained for the generaleed variational principles of plate bending problems.     

CrMnFeCoNi纳米晶温度相关的拉伸行为研究

摘要：高熵合金是一种由多种主元元素组成的新型合金。实验研究表明等原子比CrMnFeCoNi高熵合金在低温下具有比室温更高的拉伸强度和断裂韧性。本文针对这一现象，利用分子动力学模拟对平均晶粒尺寸为6 nm的CrMnFeCoNi纳米晶在300、200和77 K下分别进行拉伸模拟。模拟研究揭示了纳米尺度CrMnFeCoNi高熵合金力学行为的温度效应和强韧机理。微结构演化分析表明：低温下，塑性变形阶段，滑移系开动的较少，位错滑移所受的阻力越大，屈服强度和抗拉强度越大；模型破坏时，孔洞缺陷形核较慢，更多孔洞缺陷演化成断口，更多的断口分摊拉伸应变，使得高熵合金纳米晶的低温韧性更好。     

板弯曲与平面弹性问题的多类变量变分原理

进一步完善板弯曲与平面弹性问题的多类变量变分原理,给出了相关边界积分项的具体表达式．多类交量变分原理涵盖了平衡、应力函数、应力、位移一应变、协调和物性共五大类基本方程和所有边界条件,是一个具有更加广泛意义的变分原理．     

A method for establishing generalized variational principle

A method for establishing generalized variational principle is proposed in this paper. It is based on the analysis of mechanical meaning and it can be applied to problems in which the variational principles are needed but no corresponding variational principle is available. In this paper, the Hu-Washizu ’s generalized variational principle and the Hu’s generalized principle of complementary energy are derived from the mechanical meaning instead of from the generalization of the principle of minimum potenlial energy and the correct proofs of these two generaleed variational principles are given. It is also proved that this is wrong if one beleives that σ_ij, e_ij and ui are independent variables each other based on the reason that these three kinds of variables are all contained in these two generalized variational principles. The condition of using these two variational principles in a correct manner is also explained.