弹性大变形问题中的应力状态描述、奇异性问题和余能原理  

STRESSES,SINGULARITIES,AND A COMPLEMENTARY ENERGY PRINCIPLE FOR LARGE STRAIN ELASTICITY

作　　者：高玉臣[1] 金明[1] 兑关锁[1] 朱玉萍[1] 王志乔[1] 

机构地区：[1]北京交通大学土建学院力学所,北京100044

出　　处：《力学进展》2011年第1期93-113,共21页Advances in Mechanics

基　　金：国家自然科学基金(90205007);教育部博士点基金(20030004003);国家973项目(2010CB7321004);中国科学院非线性力学国家重点实验室开放课题(LNM201003);"北京市固体力学重点学科建设"项目资助

摘　　要：对弹性大变形理论中的3方面问题进行了综述.首先,对各种应变度量的共轭应力进行综述.大变形问题引起的应力状态描述的复杂性引起了许多学者的兴趣,对这个问题的研究也促进了大变形弹性理论的发展.在各种特定问题中,人们提出了不同的应力张量米描述应力状态,如Cauchy应力张量、第一类和第二类Piola-Kirchhoff应力张量以及Jaumann应力张量等.最近,第一作者提出了"基面力"的概念,可以描述不同应力张量的一些共同本质特征;和其它应力张量相比,用"基面力"描述应力状态更清晰、更容易理解.以"基面力"为出发点,对应力状态的各种描述进行了讨论和比较,给出了在各种边界条件下,用"基面力"表示的控制方程.其次,综述了大变形弹性力学中一些奇异性问题,包括橡胶类材料的裂纹和缺口尖端的应力奇异性、集中力作用点处的应力奇异性和楔体接触的奇异性问题.介绍用"基面力"的概念求解应力奇异性指数的方法,并和已有的研究结果进行比较.最后,回顾大变形弹性问题中余能原理的研究进展.此问题已经困扰着人们近百年.介绍了用"基面力"所描述的一种余能原理.This article reviews three aspects of large strain elasticity. First, various conjugate stress tensors to strain tensors are reviewed. Many researchers have studied the theory of large strain elasticity. Many stress tensors including the Cauchy stress tensor, the first and the second Piola-Kirchhoff stress tensor, and the Jaumann stress tensor have been proposed to describe the stress state at a point. Recently, the first author of this article proposed a concept of ＂base forces＂ to reveal the essence of stress state. By the concept of base forces, the description of the stress state becomes clearer than other stress tensors. We attempt to take base forces as a basic point of view to deal with a review in which different descriptions of stress state are discussed and compared. The governing equations and boundary conditions expressed by the base forces are given. Second, this article reviews the solution of some singularity problems for large strain elasticity, i.e., problems of stress singularity at a crack or a notch tip, at the point of application of a concentrated force and at the vertex of contact in rubberlike materials. Methods of getting the singularity index of stress by using base forces are introduced and compared to earlier work. Complementary energy principles for large strain elasticity have eluded researchers for nearly 100 years. A review of some important advances in this is also given, and a new complementary energy principle related to base forces is introduced.

关 键 词：大变形弹性理论 基面力 奇异性问题 余能原理 

分 类 号：O302[理学—力学]