模拟三维裂纹问题的扩展有限元法

›› 2010, Vol. 31 ›› Issue (10): 3280-3285.

模拟三维裂纹问题的扩展有限元法

余天堂

河海大学力学与材料学院工程力学系，南京 210098

收稿日期:2009-04-07 出版日期:2010-10-10 发布日期:2010-10-14
作者简介:余天堂，男，1971年生，博士，教授，主要从事工程力学方面的研究。
基金资助:
国家自然科学基金项目（No. 50609004，50539030）；973计划（No. 2007CB714104）。

Extended finite element method for modeling three-dimensional crack problems

YU Tian-tang

Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, China

Received:2009-04-07 Online:2010-10-10 Published:2010-10-14

摘要/Abstract

摘要：

扩展有限元法是一种在常规有限元框架内求解强和弱不连续问题的新型数值方法，其计算网格与不连续面相互独立，因此模拟移动不连续面时无需对网格进行重新剖分。给出了模拟三维裂纹问题的扩展有限元法。在常规有限元位移模式中，基于单位分解的思想加进一个阶跃函数和二维渐近裂尖位移场，反映裂纹处位移的不连续性。用两个水平集函数表示裂纹。采用线性互补法求解裂纹面非线性接触条件，不需要迭代，提高了计算效率。采用两点位移外推法计算裂纹前缘应力强度因子。给出了3个三维弹性静力问题算例，其结果显示了所提方法能获得高精度的应力强度因子，并能有效地处理裂纹面间的接触问题，同时表明扩展有限元结合线性互补法求解不连续问题具有较好的前景。

关键词: 扩展有限单元法, 三维裂纹, 接触条件, 线性互补法, 位移外推法, 应力强度因子

Abstract:

The extended finite element method (XFEM) is a new numerical method for modeling strong as well as weak discontinuities within a standard finite element framework. The computation mesh in XFEM is independent of the discontinuities, such that re-meshing for moving discontinuities can be overcome. An extended finite element method for modeling three-dimensional crack problems is described. In order to model the crack discontinuity, a Heaviside step function and the two-dimensional asymptotic crack-tip displacement fields are added to the finite element approximation for the local enrichment by using the framework of partition of unity. The crack is described with two level set functions. The nonlinear contact conditions between the crack faces are resolved with the linear complementary method; the iterative procedure is avoided. Stress intensity factors of crack front are obtained with two-point displacement extrapolation method. Three examples for three-dimensional elastostatic problems are given; the results show that the method can obtain high accurate stress intensity factors and effectively treat the contact problem between crack surfaces, and the combination of the XFEM and linear complementary method has wonderful practical merits for solving discontinuous problems.

Key words: extended finite element method, three-dimensional cracks, contact conditions, linear complementary method, displacement extrapolation method, stress intensity factors

中图分类号:

TU 443

余天堂. 模拟三维裂纹问题的扩展有限元法[J]. , 2010, 31(10): 3280-3285.

YU Tian-tang. Extended finite element method for modeling three-dimensional crack problems[J]. , 2010, 31(10): 3280-3285.

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