弹塑性裂纹和界面疲劳分析的扩展边界元法研究

The aim of this proposal is to establish a novel method named extended boundary element method (BEM) which is applied to determining the singular stress field and fatigue life of the 2-D and 3-D structures including the interfaces and cracks. The local regions around the crack tips are considered as the hardening material and in the plastic deformation. It is well known that there exists the stress singularity in the tip region. Both the non-linear governing equations and stress singularity in the plastic tip region are too difficult for conventional BEM to simulate the interface and crack propagation as plastic model. Strategy of the extended BEM is from a reformation of conventional BE analysis for the interface and crack structures. Firstly a small kernel region around the crack tip is divided from the interface/crack structure. According to the hypothesis of asymptotic stress field near the tip, the plastic stress singular orders and the associated displacement/stress eigenfunctions in the tip region can be solved by the 'total strain' theory instead of the BEM. Then the combination of the multiple eigenpairs obtained by the eigenanalysis for the tip region and the conventional BE analysis for the remaining part of the structure can result in both the singular plastic stress field near the tip and the elastic stress field of the remaining structure. Finally, based on the results of the stress and displacement distributions in the tip region and the fracture criterion, the interface and crack propagation and the fatigue life of the structure can be simulated by using the extended BEM. As a new method, however, there exist many difficulties to be overcome in the theory of the extended boundary element method and its implementation. Thus the present proposal makes a series of research tasks for solving these problems as follows: the non-linear eigenanalysis for the multiple stress singularities in the interface and crack tip region as the plastic model, regularization analysis of the nearly singular integrals in the high order elements of 3-D BEM, the technique of self-adaptive boundary element scheme on the crack surface with the crack propagation, etc. In fact, some results of the above research items can also support the applications of the extended finite element method and general BEM. After the objective of the present proposal will be finished, the extended boundary element method is expected to be an accurate and effective tool for modeling the elastoplastic interface/crack propagation and the fatigue life of the structures.

对二维和三维含裂纹和结合材料界面结构，提出和建立界面端/裂纹尖端局部塑性区域渐近应力场特征分析与边界元分析联合的扩展边界元法理论，模拟结构界面端/裂纹的疲劳扩展过程。扩展边界元法变革了断裂力学中传统边界元分析思路，将裂尖端局部塑性区按照塑性全量理论先由渐近塑性应力特征场非线性分析完成，并采纳多项的塑性应力奇异指数和相应位移/应力特征函数序列解。然后在裂尖核心区外围的弹性区用边界元离散，避免了边界元法在裂尖塑性区面临非线性和应力奇异性的双重困难。对扩展边界元法理论和实施过程中遇到的关键技术展开研究，二维和三维界面/裂尖端局部塑性区域渐近应力场多重奇异性的非线性特征分析，三维高阶边界单元几乎奇异积分的正则化算法，裂纹面扩展过程中新裂纹面边界单元自适应增加技术等研究。其中一些关键技术的获解也可供扩展有限元和其他边界元法应用。研究目标是为弹塑性断裂力学分析建立一个新的有效途径。

扩展边界元法求解断裂力学问题时，需要裂尖处应力奇异指数和相应特征函数系列解。本研究对二维和三维柱状界面、V形切口和裂纹尖端应力奇异性问题，将尖端区域位移场渐近展开成径向变量r的级数形式，引入到弹性理论/塑性理论控制方程，从而转化成线性/非线性常微分方程组特征值问题。然后采用插值矩阵法迭代求解所建立的方程组，获得了切口和裂纹尖端多重的线弹性/塑性应力奇性指数和相应的特征函数。并应用其求解了平面和反平面塑性材料切口和裂纹应力奇异性解的算例。其次，对二维和三维边界元法中采用的高阶单元，建立了高阶单元上几乎奇异积分的半解析算法，成功地计算出高阶单元上的几乎强奇异和超奇异积分，使得边界元法可以分析三维薄体结构。. 提出和建立了扩展边界元法分析弹性力学切口/裂纹结构的奇异应力场和裂纹扩展。扩展边界元法基于切口/裂纹尖端区域应力场由渐近级数表达，将裂纹尖端区域挖出后的剩余结构由边界元法分析，解出尖端区域应力场渐近级数展开项前几阶主导的幅值系数，从而获得了切口/裂纹尖端区域及结构的完整位移场和应力场，避免了常规数值方法在裂纹尖端划分高密度细网格。进一步发展了分析塑性平面切口和裂纹结构塑性奇异应力场的扩展边界元法，可以一次性计算出塑性V形切口/裂纹多重的广义应力强度因子，获得了裂尖附近的塑性奇异应力场和位移场。研究结果表明，线弹性理论和幂硬化材料塑性理论求出的尖端附近塑性区域形状是不同的，常规线弹性理论求出的塑性区类似腰形，我们按照幂硬化材料塑性理论求出的尖端塑性区类似心形。然后基于得到的裂尖附近应力场解，采用最大周向应力断裂准则，获得裂纹启裂方向和疲劳扩展路径，并研制出扩展边界元法模拟结构裂纹扩展过程自适应程序。通过算例证明扩展边界元法模拟裂纹扩展路径与实验结果吻合，表明了扩展边界元法分析裂纹疲劳扩展的有效性和准确性，具有工程应用价值。