三维切口完整奇异应力场的等几何边界元法分析研究

There widely exist notches in all kinds of engineering structures. The singular stress field at the notch root plays an important role in controlling the structure strength. The analysis of the complete singular stress field of three-dimensional (3D) notch is a key issue for ensuring the structural safety. Firstly, the integral elements in the boundary integral equation are going to be directly determined from the geometry parameter in computer aided design. The isogeometric boundary element method will be established by taking the non-uniform rational B-splines as the interpolating function, which will provide an effective numerical tool for analyzing 3D complex structure. The eigen-analysis method will be then carried out to analyze the complete stress singularity state near notch root. In the respect of choosing mathematic model, both exponential type and logarithmic type stress singularity are going to be considered. In the respect of choosing mechanical model, the opening type, sliding type, tearing type and out-plane type stress singularity will be all taken into consideration. After the eigen-equations with respect to all kinds of singularities being established, the interpolating matrix method is going to be studied to solve them for recognizing the singular characteristic state of the notch. The united program for the asymptotic expansion near notch root and the isogeometric boundary integral equation built on the region far from notch root will be proposed. The complete singularity stress field near 3D notch root is going to be semi-analytical evaluated. Basing on the computational results, the influence factors and controlling region of singularity stress are going to be investigated, and the strength criterion of 3D singularity line notch will be studied. The proposed research results will provide the guideline for designing the tolerance of structures with notches and evaluating their safety.

切口广泛存在于工程结构中，其根部的奇异应力场对结构强度起控制作用，三维切口完整奇异应力场分析是保障结构安全的关键。项目首先利用计算机辅助设计的几何建模参数直接生成边界积分单元，开展基于非均匀有理B样条插值的边界积分方程研究，建立等几何边界元法，为分析三维复杂结构锻造数值工具。随后对切口根部的完整奇异状态进行特征分析，在数学模型采用时，考虑指数型和对数型两种奇性渐近展开模式；在力学模型选用上，综合考虑张开型、滑开型、撕开型以及面外型应力奇异，建立反映切口各类奇异性的特征微分方程，研究插值矩阵法进行求解以厘清切口的奇异本征状态。最后探索切口根部奇异渐近场和远场无奇异区域上等几何边界积分方程的联立和求解方案，实现三维切口完整奇异应力场的半解析计算，继而分析奇异应力的影响因素和作用区域，开展奇异线切口断裂强度准则研究，为含切口构件容限设计及安全评价提供理论基础。

三维切口根部存在严重的奇异应力场，容易诱发裂纹，影响结构的健康与安全，需要对三维切口根部的奇异应力场进行精确计算并对含切口结构的强度进行评价。项目基于等几何边界元法来研究三维切口强度，经过四年的探索，完成了既定的研究内容。首先建立了等几何边界元分析方法，将处理Lagrange单元中几乎奇异积分的半解析方法，延伸至以非均匀有理B样条单元为基础的等几何边界元法，攻克了等几何边界元法中几乎奇异积分的计算难题。接着建立起了反应三维切口根部完整奇性状态的特征方程，探索了其数值求解方法，获取了三维切口的各类应力奇性本征状态。研究发现V形切口板在平面剪切荷载作用下，切口根部会产生一个耦合的面外奇异应力场，面外奇异模式随着切口板厚度尺寸的增大而越发重要。随后建立了三维切口根部奇异圆棱柱与远场无奇异区域上等几何边界积分方程的联合方案，在奇异区域引入渐近展开来描述物理场，剩余部分的边界积分方程采用非有理样条插值单元离散，并在等几何框架下求解，实现了对三维切口根部奇异应力场的半解析计算，可精确描述从切口尖端区域到整体结构区域的完整应力场。得益于等几何分析，可以运用较少的非均匀有理B样条单元便可以获得满意的应力计算结果。同时，在切口尖端两侧布置两个由渐近展开级数项描述的非等参元，攻克其中的主值奇异积分计算难题，构造了边界元切口奇异元，不仅可以精确求解奇异点附近的奇异应力场，还可以直接获得切口的应力强度因子。最后开展三维切口根部奇异应力的影响范围研究，建立了切口断裂的强度条件。项目研究成果不仅丰富了边界元法理论，同时可为三维含切口结构的强度设计和安全评估提供理论指导。