Note on the critical variational state in elasticity theory

Recently Prof. Chien Wei-zang pointed out that in certain cases, by means of ordinary Lagrange multiplier method, some of undetermined Lagrange multipliers may turn out to be zero during variation. This is a critical state of variation. In this critical state, the corresponding variational constraint can not be eliminated by means of simple Lagrange multiplier method. This is indeed the case when one tries to eliminate the constraint condition of strain-stress relation in variational principle of minimum complementary energy by the method of Lagrange multiplier.By means of Lagrange multiplier method, one can only derive, from minimum complementary energy principle, the Hellinger-Reissner Principle, in which only two type of in-dependent variables, stresses and displacements, exist in the new functional. Hence Prof. Chien introduced the high-order Lagrang multiplier method bu adding the quadratic terms.to original functions. The purpose of this paper is to show that by adding to original functionals one     

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.     

Semi-inverse method and generalized variational principles with multi-variables in elasticity

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On the method of Lagrange multiplier and others

The fundamentals for the correct use of the method of Lagrange multiplier are presented and illustrated by examples. Some misunderstandings of the method are clarified. Equivalent variational principles are defined. It is pointed out that for a given problem of mechanics, there may be many equivalent and unequivalent variational principles. The functional of the so called generalized variational principles of elasticity with more general forms^[16] are linear combinations of the well known functionals given by Reissner and Hu-Washizu.     

Variational principles and generalized variational principles for nonlinear elasticity with finite displacement

In a previous paper (1979)^[1], the minimum potential energy principle and stationary complementary energy principle for nonlinear elasticity with finite displacement, together with various complete and incomplete generalized principles were studied. However, the statements and proofs of these principles were not so clearly stated about their constraint conditions and their Euler equations. In somecases, the Euler equations have been mistaken as constraint conditions. For example, the stress displacement relation should be considered as Euler equation in complementary energy principle but have been mistaken as constraint conditions in variation. That is to say, in the above mentioned paper, the number of constraint conditions exceeds the necessary requirement. Furthermore, in all these variational principles, the stress-strain relation never participate in the variation process as constraints, i.e., they may act as a constraint in the sense that, after the set of Euler equations is solved, the stress-strain relation may be used to derive the stresses from known strains, or to derive the strains from known stresses. This point was not clearly mentioned in the previous paper (1979)^[1]. In this paper, the high order Lagrange multiplier method (1983)^[2] is used to construct the corresponding generalized variational principle in more general form. Throughout this paper, V/.V. Novozhilov's results (1958)^[3] for nonlinear elasticity are used.     

弹性理论中的临界变分及消除方法

临界变分现象是拉氏乘子法的固有特性，钱伟长应用高阶拉氏乘子消除了临界变分现象。本文将提出一种新的方法－凑合反推法，这种方法摒充了拉氏乘子法，把拉氏乘子所在的项目一个待定函数Ｆ代替。这样构成的泛函，作者称之为试泛函。而待定函数Ｆ的识别类似于拉氏乘子的识别。通过该法可以方便地构造出各种多变量广义变分原理，并且可以消除临界变分现象。     

Generalized variational principles with several arbitrary parameters and the variable substitution and multiplier method

The functional transformations of variational principles in elasticity are classified as three patterns: Ⅰ relaxation pattern, Ⅱ augmented pattern and III equivalent pattern.On the basis of pattern Ⅲ, the generalized variational principles with several arbitrary parameters are formulated and their functionals are defined. They are: the generalized principle of single variable u with several parameters, the generalized principle of two variables u, σ with several parameters, the generalized principle of two variables u, ε with several parameters, and the generalized principle of three veriables u, ε, σ with several parameters. From these principles, a series of new forms of equivalent functionals can be obtained. When the values of these parameters are properly chosen, a series of finite element models can be formulated.In this paper, the question of losing effectiveness for Lagrange multiplier method is also discussed. In order to "recover" effectiveness for multiplier method, a modified method, namely, the variable substitution and multiplier method, is proposed.     

刚塑性广义变分不等原理及其在平面应变分析中的应用

首先利用Lagrangian乘子法，从势能角度出发构造了考虑摩擦效应这一能导致变分不等形式的广义能量泛函，把一般的有条件的变分原理化为无条件的变分原理唯一确定，得出了各Lagrangian乘子所代表的物理意义。建立了刚塑性理论中的Coulomb摩擦约束的广义变分不等原理。而后基于退化的摩擦约束广义变分等式原理，对长矩形板镦粗进行了塑性加工工步分析，所得结果与经典上限法结果相吻合。     

Classification of variational principles in elasticity

In this paper, variational principels in elasticity are classified according to the differences in the constraints used in these principles. It is shown in a previous paper [4] that the stress-strain relations are the constraint conditions in all these variational principles, and cannot be removed by the method of linear Lagrange multiplier. The other possible constraints are four of them: (1) equations of equilibrium, (2) strain-displacement relations, (3) boundary conditions of given external forces and (4) boundary conditions of given boundary displacements. In variational principles of elasticity, some of them have only one kind of such constraints, some have two kinds or three kinds of constraints and at the most four kinds of constraints. Thus, we have altogether 15 kinds of possible variational principles. However, for every possible variational principle, either the strain energy density or the complementary energy density may be used. Hence, there are altogether 30 classes of functional of variational principles in elasticity. In this paper, all these functionals are tabulated in detail.     

论拉氏乘子法及其唯一性问题

本文指出文[13](1985)对于拉氏乘子法的最近论点仍旧是先验的,并不是国际上大家所公认而又证实了的“古老的数学概念”(1983),该文所赖以立论的三个实例,都不成立。所说明的,不是象文中所称的那样,“在力学问题中正确应用拉氏乘子法的要点”,恰好相反,文[13]很不正确地应用了拉氏乘子法,从而达到了错误结论,甚至只能求助于所谓“猜谜语“的方法。 本文也指出拉氏乘于是可以根据拉氏乘子法唯一地识別的,文[10]、文[16]说拉氏乘子的不唯一性应是对拉氏乘子法的误解所引起的。 本文讨论的弹性力学问题是非线性弹性体的一般弹性力学问题,其应力应变关系是非线性的,当应变很小可以略去其非线性项时,其结果可以还原为线性弹性体的各种广义变分原理。因此,不论Hellinger-Reissner 原理或胡-鹫原理都是本文所讨论的非线性弹性体的广义变分原理的近似特例。     

Variational Principles for EIastoplastic Continua∗

ABSTRACT

A mixed variational principle is constrained by a homogeneous yield function using a Lagrange multiplier. The Lagrange factor corresponds to the scalar factor in Prager's normality rule for the plastic strain increments. Several reduced functionals and their associated constitutive equations are derived by eliminating some variables.     

Further study of the equivalent theorem of Hellinger-Reissner and Hu-Washizu variational principles

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COUPLED VARIATIONAL PRINCIPLES AND GENERALIZED COUPLED VARIATIONAL PRINCIPLES IN PHOTOELASTICITY

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The Lagrange Multiplier in Incompressible Elasticity Theory

The first variation condition for the potential energy in nonlinear elasticity for incompressible materials provides a linear functional which vanishes on an appropriately constrained set of variations. We prove a representation theorem for such linear functionals which forms the basis for the existence of a constraint reaction (Lagrange multiplier) field. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the canonical equations of Kirchhoff-Love theory of shells

The paper outlines a procedure to derive the canonical system of equations of the classical theory of thin shells using Reissner’s variational principle and partial variational principles. The Hamiltonian form of the Reissner functional is obtained using Lagrange multipliers to include the kinematical conditions that follow from the Kirchhoff-Love hypotheses. It is shown that the canonical system of equations can be represented in three different forms: one conventional form (five equilibrium equations) and two forms that are equivalent to it. This can be proved by reducing them to the same system of three equations. For problems with separable active and passive variables, partial variational principles are formulated __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 10, pp. 99–107, October 2007.     

摩擦约束塑性力学变分不等原理的半反推法

带摩擦约束的弹塑性接触问题,由于摩擦约束条件是一种判别性的条件,它的变分问题的逆问题的研究比较困难。本文对弹塑性接触力学中的变分不等问题的逆问题进行了研究,改进了半反推法并将其应用到弹塑性变分不等原理的研究中,导出了摩擦约束弹塑性增量广义变分不等原理中的能量泛函,消除了用拉氏乘子法可能产生的临界变分现象,在证明中,巧妙地处理了增量表示的接触摩擦边界条件,避免了使用非线性泛函分析和凸分析,简化了证明。     

超细长弹性杆的分析力学问题

超细长弹性杆作为DNA等生物大分子链的力学模型，其平衡和稳定性问题已成为力学与分子生物学交叉的研究热点．虽然在Kirchhoff动力学比拟的基础上，用分析力学方法讨论弹性杆的文章已见诸文献，但尚未形成弹性杆分析力学的严格理论．本文研究了超细长弹性杆分析力学的若干基础性问题．对杆截面的自由度、虚位移、约束方程及约束力等基本概念给出严格的定义和表达式．建立弹性杆平衡的D’Alembert-Lagrange原理、Jourdain原理和Gauss原理；从D’Alembert-Lagrange原理导出Hamilton原理．从变分原理出发导出Lagrange方程、Nielsen方程、Appell方程和Hamilton正则方程；对于受约束的弹性杆，导出了带乘子的Lagrange方程．讨论了Lagrange方程的首次积分．对于杆中心线存在尖点的情形，导出了微段杆平衡的近似方程。     

Variational Formulation of the First Principle of Continuum Thermodynamics

The First Principle of Continuum Thermodynamics is formulated as a variational condition whose test fields are piecewise constant virtual temperatures. Lagrange multipliers theorem is applied to relax the constraint of piecewise constancy of test fields. This provides the existence of square summable vector fields of heat flow through the body fulfilling a virtual thermal work principle, analogous to the virtual work principle in Mechanics. The issue of compatibility of thermal gradients is dealt with and expressed by the complementary variational condition. Primal, complementary and mixed variational inequalities leading to computational methods in heat-conduction boundary-value problems are briefly discussed.     

不可压饱和粘弹性多孔介质固结问题的Gurtin型变分原理

基于多孔介质理论,在固相骨架和孔隙流体微观不可压,固相骨架小变形且满足线性粘弹性积分型本构关系的假定下,利用卷积积分的性质,本文首先建立了以固相骨架位移、孔隙流体相对速度和孔隙流体压力为宗量的流体饱和粘弹性多孔介质固结问题的一个Gurtin型变分原理.其次,利用Lagrange乘子法解除相关的变分约束条件,建立了流体饱和粘弹性多孔介质固结问题的若干广义Gurtin型变分原理,包括第三类的Hu-Washizu型变分原理.最后,简单讨论了等价初边值问题的相应变分原理.这些Gurtin型变分原理的建立不仅丰富了饱和粘弹性多孔介质的相关理论,而且为相关数值模拟方法,如有限元法、无网格法等的建立奠定了理论基础.