基于本征COD边界积分方程的多裂纹固体大规模数值模拟及其力学行为研究

Newly techniques are in urgent need to rapidly and accurately analyze and solve the research on the large-scale numerical modeling for multi-cracked solids with large quantity of cracks, which is a challenging task for researchers. A series of beneficial exploration for multiple crack problems by employing the eigen crack opening displacement (COD) boundary integral equations have been investigated in this study: ① the concept of eigen COD is proposed. Based on the concept of eigen COD, the numerical models of eigen COD boundary integral equations are to be developed. ② the local Eshelby matrices, reflecting the interaction among those cracks with relatively small distances but have strong influences, are proposed by applying a superposition technique which divides all cracks into two groups, i.e., adjacent group and far-field group. The corresponding numerical iteration can be avoided divergence with the introduction of local Eshelby matrix under dense crack circumstances. ③ the numerical models of eigen COD boundary integral equations are designed and implemented in an iterative fashion. The accuracy and efficiency are verified. ④ the relations between overall properties of multi-cracked solids (such as the strengths, rigidities, anisotropies and fracture properties) and the characteristics with respect to random distribution of cracks (such as the crack numbers, positions, shapes, orientations and sizes) are to be researched, providing a reliably numerical analysis method for dealing with randomly distributed crack problems in future processing. In conclusion, not only the computational accuracy and efficiency can be guaranteed, but also the overall properties and local details can be obtained. The numerical models of eigen-COD boundary integral equations realize the numerical simulation for multiple crack problems with large quantity of cracks using ordinary desk-top computers. The research is expected to afford the references for the safety evaluation of multi-cracked solids during service. Numerical results clearly demonstrate the numerical models of eigen COD boundary integral equations have great significances both theoretically and practically.

多裂纹固体大规模数值模拟是富有挑战性的课题，如何快速准确地分析和求解是亟需的技术。本项目通过本征裂纹张开位移（COD）边界积分方程对多裂纹问题进行一系列有益的探索：①提出本征COD的概念，建立本征COD计算模型；②划分近场和远场裂纹，引入反映裂纹虚拟面力与本征COD关系的局部Eshelby矩阵，处理裂纹间的相互影响，避免数值迭代发散；③设计并实现本征COD计算模型迭代算法，验证其有效性、计算精度和效率；④研究多裂纹固体的整体性能（强度、刚度、各向异性和断裂性能）与裂纹分布的随机特性（裂纹数量、位置、形状、方位和尺寸）的关系，为处理随机分布多裂纹问题提供可靠的数值分析手段。本征COD计算模型为求解多裂纹问题提供新的能够兼顾计算精度与效率、整体性能与局部细节的数值方法，在普通计算机上实现多裂纹固体力学性能的大规模数值模拟，为多裂纹固体在使用期间的安全评价提供参考，具有重要的理论和工程应用意义。

多裂纹固体大规模数值模拟是富有挑战性的课题，如何快速准确地分析和求解是亟需的技术。本项目通过本征裂纹张开位移（COD）边界积分方程对多裂纹问题进行一系列有益的探索：①提出了本征COD的概念，建立了多裂纹问题的本征COD计算模型；②将多裂纹问题划分为近场和远场裂纹，引入反映裂纹虚拟面力与本征COD关系的局部Eshelby矩阵，处理裂纹间的相互影响，避免了数值迭代发散；③设计并实现本征COD计算模型迭代算法，验证其有效性、计算精度和效率；④建立并编制了对偶边界积分方程方法、数值格林函数法和快速多极边界元法的Fortran计算程序，通过若干数值算例，分别从计算精度和计算效率两方面来综合比较本征COD计算模型的可行性和高效性；⑤基于拉格朗日插值多项式，处理椭圆裂纹问题。利用椭圆的对称性和周期性等几何特征，采用高阶形状函数，在单元的径向和周向上重复使用实节点进行插值，实现单元内部的光滑性，消除了传统低阶单元中存在的端节点/线效应；⑥针对当前异质体定义了本征应变概念的同时，考虑边界或周边异质体的影响，结合Eshelby等效夹杂物的概念，将任意载荷作用下当前异质体的本征应变用离散的夹杂物Eshelby矩阵来描述和计算，得到了新的本征应变格式的边界积分方程计算模型，初步实现了对压电材料半平面梯形/正方形夹杂问题的数值模拟。