Numerical Modeling for the Analysis of Crack-Inclusion Problems by the Eigen Iterative Computational Model
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摘要: 针对固体材料含有“裂纹-夹杂”的数值模拟问题，将Eshelby本征应变和等效夹杂替换理论引入边界积分方程中，建立了本征裂纹张开位移(crack opening displacement, COD)和本征应变边界积分方程的计算模型及数值实现，以解决“裂纹-夹杂”的相互作用机制. 在一定条件下，异性夹杂可以作为一般的夹杂问题处理，物理上可令夹杂的弹性模量为零，则该夹杂就“退化”为孔洞；同时，在几何上可令其最小尺寸方向上的尺寸为零，即可进一步“退化”为裂纹，因而裂纹可被认为是弹性模量为零的一种特殊夹杂. 采用边界积分方程的离散形式对裂纹和夹杂问题进行了数值验证，其中裂纹和夹杂的边界分别采用Gauss配点法和边界点法进行离散，进行了应力分析，研究了裂纹与夹杂的相互作用. 数值算例验证了本征计算模型处理“裂纹-夹杂”问题的正确性和方法的可行性，也表现出较高的计算精度，为该计算模型的大规模数值分析奠定了理论基础.Abstract: For the numerical modeling of solid materials containing crack-inclusions, the theory of Eshelby's eigenstrain and equivalent inclusion was incorporated into the boundary integral equations (BIEs). A computation model and an iterative algorithm for the eigen crack opening displacement (COD) and eigenstrain BIEs to address the interaction of crack-inclusions were proposed. Under certain conditions, anisotropic inclusions may be considered as general inclusions. When the elastic modulus of the inclusion is set to zero, the inclusion will degenerate into a hole, and geometrically, with its minimum dimension reduced to zero further it will degenerate into a crack. Thereby, a crack was classified as a special type of inclusion with zero elastic modulus. The discrete form of the boundary integral equation was utilized for the numerical validation of cracks and inclusions, with their boundaries discretized with Gaussian integration points and the boundary point methods, respectively, to conduct stress analysis and investigate the interaction between crack-inclusions. Numerical examples confirm the correctness and feasibility of the eigen iterative model in modeling crack-inclusion problems, demonstrating its high computation accuracy, and providing the theoretical foundation for future large-scale numerical analysis with this computation model.
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Key words:
- crack-inclusion problem /
- eigen COD /
- eigen strain /
- boundary integral equation /
- numerical modeling
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图 1 本征COD与本征应变的关系
Figure 1. The relationship between the eigen COD and the eigen strain
图 2 本征计算模型基本流程图
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 2. The flowchart of the eigen iterative computation model
图 4 一条水平裂纹和一椭圆夹杂问题的Ⅰ型裂纹应力强度因子比较
Figure 4. Comparisons of normalized KⅠ SIFs for one horizontal crack and one elliptical inclusion
图 5 一条水平裂纹和一椭圆夹杂问题的Ⅱ型裂纹应力强度因子比较
Figure 5. Comparisons of normalized KⅡ SIFs for one horizontal crack and one elliptical inclusion
图 6 一条水平裂纹和多椭圆夹杂问题
Figure 6. One horizontal crack surrounded by a square array of 8 elliptical inclusions
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