【作者】 孔凡忠；

【导师】 姚振汉； 郑小平；

【作者基本信息】 清华大学 ， 固体力学， 2001， 博士

【摘要】 随着科技进步和现代工业的发展,各种复合材料在工程中得到了广泛应用。因此,复合材料宏观等效力学特性的研究受到了学术界和工程界的共同关注。复合材料通常可以看作在基体材料中嵌入了各种不同的夹杂相,其宏观等效力学特性主要取决于所嵌入夹杂相的尺寸、形状、性质、体积比和空间分布。因此,含随机分布多种夹杂相的固体可以作为这类复合材料的力学分析模型。 对于研究不同夹杂相所带来的非均匀性问题,边界元分域解法是一种可行的数值分析方法,它比有限元法等其他数值方法具有更高的计算精度。对于含随机分布n个夹杂相的固体而言,采用常规的边界元分域解法进行计算,我们将会得到一个n+1子域问题,如果直接进行计算,计算复杂性将随着夹杂数目的增加而急剧增加。 为了克服上述困难,根据各个夹杂相积分区域的相似性,本文提出了边界元相似子域法。把含随机分布多种夹杂相的固体归结为含内边界条件的复连通域问题来求解,从而极大地提高了计算效率。大量数值算例表明,与有限元法相比,本文提出的边界元相似子域法具有更高的计算精度和计算效率,因此更加适合于复合材料宏观等效力学特性的数值模拟。 利用边界元相似子域法,本文对各种二维问题进行了计算,其中包括:含随机分布圆形夹杂的平面应力问题(颗粒增强复合材料)、含随机分布椭圆形夹杂的平面应变问题(长纤维增强复合材料)和含随机分布多种夹杂相的薄板。这些计算为相应复合材料宏观等效力学特性研究提供了可靠的数值模拟方法。 文中以含100个随机分布圆形夹杂的固体板材为例,利用边界元相似子域法进行了大量数值计算,得到了宏观等效力学特性的边界元数值解,并以此为依据,对“等效介质近似方法”的各种经典近似解法进行了综合分析,得到了一些具有重要参考价值的结论。 本文提出的边界元相似子域法也可以推广应用于夹杂相外缘附有界面层结构的复合材料的数值模拟。与边界元多极快速算法相结合,还可以推广应用于对复合材料进行三维数值模拟,具有广泛的应用前景。更多还原

【Abstract】 Along with the development of modern industry and technology, various composite materials are increasingly applied in engineering projects. Therefore, the investigation of the macroscopically effective mechanical properties of the composite materials attracts much attention of both researchers and engineers. Generally, the composite materials can be regarded as a matrix material with plenty of separated inclusions. Their macroscopically effective mechanical properties strongly depend on the sizes, shapes, properties, volume fraction and spatial distribution of the inclusion phases. Consequently the solids with randomly distributed inclusions of various shapes, sizes and materials can be regarded as the mechanical model of such composite materials.For the investigation of the heterogeneity due to different inclusion phases, multi-region BEM can be regarded as a feasible numerical method, which is more accurate than FEM and other numerical methods. As for the solids with n randomly distributed inclusion phases, if the conventional multi-region BEM is adopted, the equation system for n + \ subdomains should be formulated. If such equation system is solved directly, the computational complexity will increase significantly as the number of inclusions increases.In order to overcome the above-mentioned difficulty, similar subdomain BEM scheme is presented in this paper, based on the similarity of the integral area of inclusion phases. The solid with randomly distributed various inclusions can be reduced to a multiply connected domain of the matrix with inner boundary conditions. In this way, the computational efficiency is enhanced significantly. A lot of numerical examples indicate that the presented similar subdomain BEM has a higher computational accuracy and computational efficiency than FEM, so it is more suitable for the numerical simulation of the macroscopically effective mechanical properties of composite materials.Using the similar subdomain BEM, various 2D problems have been computed in this paper, including: the plane stress problem with randomly distributed circularinclusions (granular reinforced composite material), the plane strain problem with randomly distributed elliptical inclusions (long fibrous reinforced composite material), and thin plate with randomly distributed various inclusions. These computational schemes provide reliable numerical simulation methods for the investigation of the macroscopically effective properties of the corresponding composite materials.As numerical examples, plenty of numerical computation for the plates with 100 randomly distributed circular inclusions is carried out by the similar subdomain BEM, and the numerical solution of the macroscopically effective mechanical properties have been obtained. Based on the above numerical solution, the classical approximate solutions of the effective medium approximation have been analyzed in detail and some important referential conclusions have been obtained.The similar subdomain BEM presented in this paper can also be successfully applied to the numerical simulation of the composite materials with different interphase layer. On the other hand, similar subdoamin BEM can be generalized to the numerical simulation of 3D solids with randomly distributed inclusions combining with the fast multipole BEM scheme. In a word, similar subdomain BEM scheme has an extensive applied prospect.更多还原