摩擦约束广义变分不等原理的临界变分现象及分析  

Crisis in Generalized Variational Inequalities Principles with Frictional Constraints

作　　者：孙辉[1] 扶名福[2] 

机构地区：[1]南昌工程学院计算机科学与技术系,江西南昌330099 [2]南昌大学机电学院,江西南昌330047

出　　处：《力学季刊》2008年第1期158-165,共8页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(19762002;50539020)

摘　　要：对于具有摩擦约束的弹塑性接触问题,由于边界接触面上的摩擦力由不等式表示,导致得到包含摩擦约束的广义变分原理为广义变分不等原理。广义变分不等原理通过将摩擦力纳入问题的能量泛涵,可避免考虑摩擦力变化的具体过程,便于数值方法如有限元等在弹性接触问题上的应用。但是,通过对广义变分不等原理的研究,发现在弹性力学广义变分不等原理中,势能型和余能型广义变分不等原理,均存在临界变分现象,即变分时拉格朗日乘子为零,变分失败;或者得到的能量泛函变分后得不到问题的欧拉方程。在对弹性力学广义变分不等原理临界变分现象进行分析后,提出了避免发生临界变分现象的方法。实际应用证明了方法的有效性。通过避免临界变分现象的发生,可以保证拉格朗日乘子方法的有效使用。In the case of elastic contact problem wiht friction, the fricton law on the contact boundary is expressed by inequalities. Thus the generalized variational principles of the constains lead to generalized variational inequality principles, instead of generalized variational ones. Generalized variational inequality principles are often used to solve elastic contact problems by using numerical simulation such as finite element method in which the frictional constraints are present, for it is not necessary to deal wiht the change of the friction force. It is demonstrated in this paper that the generalized inequalities variational principles, established by Lagrange multipliers method, may result in failures or crisis, in which the Lagrange multipliers may vanish or no relevant Euler equations can be derived. After analyzing the crisis phenomenon, a method was then proposed to overcome this kind of failure or crisis. The experimental result indicates that avoiding the variational crisis may validate the Lagrange multiplier method.

关 键 词：摩擦约束 变分不等原理 弹性力学 拉氏乘子法 

分 类 号：O343.2[理学—固体力学]