多尺度多裂纹固体的本征COD边界积分方程数值模拟

The local Eshelby matrices, reflecting the interaction among cracks, are to be derived with the aid of investigating the integral interpolation techniques and the numerical representation of eigen crack opening displacement (eigen-COD) over three-dimensional crack surfaces. In what follows, the numerical models of eigen-COD boundary integral equations are to be developed with the corresponding numerical solution procedures in an iterative manner to realize the large-scale numerical simulations for multi-scaled multi-cracked solids with large quantity of cracks using ordinary desk-top computers. There are two problems to be resolved with great significances both theoretically and practically through the analysis and computations: (1) the properties of solids such as the strengths, rigidities, Poisson ratios, anisotropies, fracture properties correlated to the crack information including the crack numbers, positions, shapes, orientations and size distributions in solids; (2) the relations between the reasonable choice of the representative volume element (RVM) for the multi-scaled multi-cracked solids and the characteristics of materials with respect to the hierarchically multi-scaled structures. There are two distinct features being very favorable in the eigen-COD boundary integral equation models: (1) both the overall properties and local details of the RVE can be obtained; (2) both the accuracy and efficiency of computation can be guaranteed. The research is expected to enhance the development in the large scale numerical simulation for multi-scaled multi-cracked solids, and afford the references for the safety evaluation of elements and devices made of solids with cracks during service, and provide the experiences for the simulation of corresponding physical properties, and enhance the advances in the computational materials sciences and their application in engineering.

通过研究三维裂纹表面的积分插值技术和本征裂纹张开位移（本征COD）的数值表征，推导出反映裂纹之间相互作用的局部Eshelby矩阵；以此为基础，建立本征COD边界积分方程的数值模型，发展相应的数值迭代解法，在普通台式机上实现多尺度多裂纹固体的力学性能的大规模数值模拟。通过分析与计算解决两个具有重要理论与实际意义的问题：（1）强度、刚度、泊松比、各向异性以及断裂性能与固体中裂纹的数量、位置、形状、方位和尺寸分布的关系；（2）代表性体积单元（RVE）的选择与材料结构多层次多尺度特性的关系。本征COD边界积分方程模型具有的两个鲜明特色是：（1）能够兼顾研究对象的整体性能与局部细节；（2）能够体现计算精度与效率的统一。该项研究将推动多尺度多裂纹固体大规模数值模拟领域的研究进展，为含裂纹固体元器件在服役期间的安全性评价提供参考，为相关物理性能的数值模拟提供借鉴，并促进相关计算材料科学的发展和工程应用。

针对含有密集裂纹和粒子固体的大规模数值模拟问题，通过近场群和远场群和划分，提出和推导了反映近场群裂纹与裂纹、粒子与粒子以及裂纹与粒子之间相互作用的局部Eshelby矩阵，开展了系列工作，建立了本征COD和本征应变边界积分方程的数值计算模型与相应的迭代算法，从而能够在普通台式机上实现多裂纹多粒子固体力学性能的大规模数值模拟。经过多种尝试，找到了解决粒子本征应变常数假定这一局限的新思路，开展了对偶边界积分方程建模的研究和相应的程序设计工作，作为建立新的本征变量（含本征COD和本征应变）边界积分方程计算模型的基础。对高次边界单元和高次裂纹单元进行了初步的研究，表明采用高次单元的离散方式能够进一步提高计算模型的精度与效率，特别对三维固体的数值模拟来说更加合理有效。本征变量边界积分方程计算模型能够兼顾研究对象的整体性能与局部细节并实现计算精度与计算效率的统一，促进相关计算材料科学的发展和工程应用。