【作者】 程长征；

【导师】 牛忠荣；

【作者基本信息】 合肥工业大学 ， 工程力学， 2007， 博士

【摘要】 本文在调查和总结现有的分析涂层结构和V形切口方法的基础上，详细研究了使用边界元法分析涂层和V形切口结构的力学场问题。创立了一个新的分析途径，研制了相应的计算程序，有效和准确地求解了涂层结构内的物理场和V形切口尖端附近的奇异应力场。全文主要研究工作及结论如下：1)研究了二维涂层结构温度场和应力场边界元法中几乎奇异积分的计算。将涂层结构分成涂层和基体两种不同的子域，在涂层域中使用完全的解析积分算法，解决了其中的几乎奇异积分难题，使边界元法可以求解超薄涂层结构中全域的温度场和应力场分布。使得边界元法可以有效分析涂层结构内的物理场，发挥了边界元法计算量小、精度高的优势。同时运用该法分析了碳纤维布加固钢结构的强度和浅表面裂纹应力强度因子等问题。2)研究了三维薄形层合结构边界元法中几乎奇异积分的半解析算法。该算法使得边界元法不仅可以计算更加靠近边界的各层内点力学参量，并且能分析层厚更薄的三维层合结构的位移场和应力场。3)研究了二维应力边界积分方程中几乎超奇异积分的降阶。通过分部积分变换消除了其中的超奇异积分，获得仅含强奇异积分的应力自然边界积分方程。对于近边界应力的计算，进一步运用正则化算法解析计算其中的几乎强奇异积分。创新地将该技术推广到热弹性力学和弹性力学多域边界元法中。较常规边界元法相比，应力自然边界积分方程可以求解离边界更加接近的内点应力值。4)首次提出边界元法计算V形切口应力奇性指数的一个新技术。基于线弹性力学理论，将切口尖端的位移和面力按级数渐近展开，代入到边界积分方程中，离散后转换成关于切口奇性指数的代数特征值问题，利用QR法求解获得V形切口的应力奇性指数。这一新方法避免了在切口尖端布置细密单元，并可同时求出多阶应力奇性指数。5)创新建立了边界元法计算V形切口奇异应力场的新途径。将含V形切口结构分成围绕切口尖端的小扇形和剩余结构两部分。基于切口尖端区域求出的多重应力奇性指数和相应的位移、应力特征角函数，将小扇形区域的位移和应力表示成有限项奇性指数和特征角函数的线性组合，代入到在挖去小扇形后的剩余结构内建立的边界积分方程。由此准确地计算出V形切口尖端区域的位移场、多重奇异应力场和应力强度因子。然后又将该法推广到粘结多材料V形切口尖端奇异应力场分析以及多重应力强度因子的计算。这一新方法完整符合了切口尖端奇异应力场的解析规律。本文结果为V形切口的疲劳、断裂分析提供了准确的应力场分布。更多还原

【Abstract】 Based on the review of the analytic methods of coating-structures and V-notched structures, boundary element analysis of the coating-structures and V-notched structures are studied in detail by the author. The main work and contribution in this thesis are given as follows:1) The evaluation of the temperature field and two-dimensional stress field in the coating-structures is studied by boundary element method. Because the thickness of the coatings is very thin, the nearly singular integrals will occur in the boundary element analysis of the coating-structures. Here, a completely analytic algorithm has been raised to deal with the nearly singular integral. Consequently, the boundary element method can be used to efficiently calculate the temperature field, the stress and displacement fields of the coating-structure with multilayer materials. Then, the present method is adopted to determine the strength of the carbon fiber reinforced structures and the stress intensity factors of the sub-surface cracks.2) The semi-analytic algorithm of the nearly singular integral is introduced to the boundary element analysis of three-dimensional thin laminated structures. By using the semi-analytic formulation, the boundary element method can not only calculate the mechanical parameter of the inner points very close to the boundary in each layer, but also analyze the stress and displacement fields of multilayer thin-walled structures.3) The evaluation of the hyper-singular integral in the stress boundary integral equations (BIE) is studied. A series of transformations are performed to the conventional displacement derivative BIE in order to eliminate the hyper-singular principal value integrals. Hence,a new stress natural BIE is developed, in which there only exist the strongly singular integrals instead of the hyper-singular integrals in the conventional stress BIE. Furthermore, when a source point tends to the boundary, the small dominant factor leading to the nearly strongly singular integrals in the natural BIE is shifted out of the integral representations by the integration by parts, so that the singular integrals are accurately calculated. As a result, the present method is extended to the thermo-elasticity and multi-domain boundary element method by the author. Numerical examples demonstrate that the natural BIE can successfully determine the stress distributions in the domain very closer to the boundary in comparison with the conventional BIE.4) A new technique about the evaluation of the stress singularity orders of the V-notches by boundary element method is firstly proposed. Based on the theory of linear elasticity, the asymptotic displacement and stress fields in the V-notch tip region are expressed as a series expansion with respect to the radial coordinate from the tip. The series expansion of the asymptotic field is then substituted into the equations of the boundary element analysis of the V-notched structure. After the discretization, the boundary integral equation is transformed to the eigen equation with the stress singularity orders. By the use of the QR method to solve the eigen equation, the eigenvalues which are the singularity orders can be obtained: Hence, the use of very fine elements near the V-notch tip in the conventional boundary element method is unnecessary in the present new method. The multiple singularity orders of the V-notch can be obtained simultaneously in the present method. 5) A new way to determinate the singularity stress field near the V-notch tip by the boundary element method is established. Firstly, the V-notched structure is divided into two parts, a small sector around the V-notch tip and the other. Based on the computed multiple stress singularity orders and the corresponding eigen functions of the displacements and stresses, the displacements and stresses in the small sector are expressed as the linear combinations of the finite terms of the series expansion with all the singularity orders. Secondly, the boundary element method is used to model the V-notched structure removed the small sector, in which the boundary conditions along the arc edge from cutting the sector are expressed by the above linear combinations. Finally, the displacement and stress field at the V-notch tip and the multiple stress intensity factors are obtained through the boundary element analysis. This new method reflects the completely analytic character of the singular stress field near the V-notch tip. The accurate stress fields obtained by the present method are very useful in the analysis of the fatigue and fracture of the V-notched structures.更多还原