【作者】 王克用；

【导师】 亢一澜； 秦庆华；

【作者基本信息】 天津大学 ， 固体力学， 2006， 博士

【摘要】 接触问题广泛存在于生产和工程实际中。待定的接触区域和摩擦效应的不可逆性使得接触问题表现出高度的非线性特征。尽管研究者们已经获得了许多接触问题的解析解,但是绝大多数的实际问题都超出了其应用范围。因此,出现了大量的求解接触问题的数值方法,这些数值方法主要是以传统有限元法和传统边界元法为基础的。作为一种高效有力的计算工具,杂交Trefftz有限元法自30年前提出以来,已经受到了愈来愈多的重视。它主要具备两个优点:其一,单元公式中只含有边界积分,这样就可以构造出任意多边形甚至曲边单元,它可以看作是一种特殊的对称形式的边界型求解方法。因此,Trefftz有限元法具备了传统边界元法的优点,而又避免了复杂奇异积分方程的计算。其二,与传统有限元法相比,Trefftz有限元法在处理带有各种因载荷、几何(如角点、孔洞、裂纹、夹杂、集中载荷等)引起的局部效应问题时,不需另外细分网格,就能得到比较理想的精度。到目前为止,Trefftz有限元法已经应用到诸多工程领域。然而,将这种有限元法应用于接触问题尚无先例。本文主要分两部分内容:对于Trefftz有限元方法本身的一些研究和基于这种有限元法开发接触问题的求解算法。从弹性平面问题的基本解析解出发,提出了建立普通单元的Trefftz函数的另一种途径。另外,在Piltner工作的基础上,通过旋转映射函数的引入以及Muskhelishvili复变函数法的应用,提出了一种新的Trefftz孔洞单元。这种新单元能够方便地分析含有任意方向椭圆孔的问题,弥补了早期Piltner孔洞单元的不足。本文首次将Trefftz有限元法应用于摩擦和无摩擦弹性接触问题。应用静凝聚将原来较大的接触模型缩聚成较小的接触模型,而只保留了可能接触区内的离散节点。缩聚后的模型降低了机时,提高了计算效率。由于接触界面方程的系数矩阵存在零主元,因此采用列主元高斯消去法进行求解。对于无摩擦情况,只需单步加载;而对于摩擦情况,则采用自动载荷增量技术进行加载。在整个应用过程中,本文提出的直接约束-Trefftz有限元法概念简单、物理意义明确、应用方便。更多还原

【Abstract】 Contact problems arise frequently in industrial processes and engineering applications. The non-linearities appearing in these problems are due to the undetermined size of contact zone and to the irreversible nature of frictional effects. Even though a number of analytical solutions are available, the vast majority of practical problems lie beyond their applicability. Hence a large family of numerical approaches, mainly based on the conventional finite element method (FEM) or boundary element method (BEM), has been developed for treating these problems. As a highly efficient and well established tool, the hybrid-Trefftz (HT) FEM, initiated about three decades ago, has now become more and more popular. There are two main advantages in HT FEM: firstly, the formulation only calls for integration along the element boundary which enables arbitrary polygonal or even curve-sided elements to be generated. As a result, it may be considered as a special, symmetric, substructure-oriented boundary solution approach and thus possesses the advantages of the conventional BEM. In contrast, however, HT FEM avoids the introduction of singular integral equations and, secondly, HT FEM, compared with the conventional FEM, can accurately treat problems with various local effects due to loading and/or geometry (for example, corners, holes, cracks, inclusions, concentrated loads, etc.) without troublesome mesh refinement. Up to now, HT FEM has been thoroughly explored in a number of fields in engineering. However, there are no contributions by HT FEM for contact problems. The main content of this paper is twofold: some research on HT FEM itself and, development of contact solution algorithm based on it. Starting from the fundamental analytical solutions to plane elastic problems, an alternative way to construct Trefftz functions for regular element is presented. Additionally, a novel Trefftz hole element, based on the work of Piltner, is developed with the introduction of rotated mapping function and with the help of Muskhelishvili complex function approach. This element can be easily and conveniently employed to analyze problems with elliptical holes opened in any direction. And thus the shortage of the Piltner hole element has been covered.This paper initially applies HT FEM, in conjunction with the direct constraint technique, to elastic contact problems with or without friction. The static condensation is employed to condense a larger model down to a smaller one which involves discrete更多还原