A NUMERICAL EQUIVALENT INCLUSION METHOD FOR DETERMINING THE INTERACTION ENERGY BETWEEN INHOMOGENEITIES AND DISLOCATIONS
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摘要: 工程结构中的夹杂物和位错会极大地影响材料的力学性能和服役寿命。以往的解析解主要关注特定形状(如圆形、椭圆形)夹杂物与位错之间的相互作用。当采用数值方法计算时，由于位错的奇异性，即使是商用有限元软件也会面临处理上的困难。该文基于数值等效夹杂算法并结合快速傅里叶变换，求解了无穷体内夹杂物与刃型位错的交互能，有效地规避了数值奇异性问题。相对误差的范数分析结果表明，在杂质附近所产生的应力扰动对最终结果具有较大影响。该文计算方法能够更加精确地确定应力扰动场，并显示出优越的数值收敛性和稳定性。在求解任意形状杂质与位错相互作用问题中，该文提供了一种便捷且有效的计算方案。Abstract: The mechanical properties and service life of materials may be significantly influenced by the interaction between inhomogeneities and dislocations in engineering structures. Previous analytical studies on inhomogeneity-dislocation interactions have been concerned primarily on some special inhomogeneity shapes (e.g. circle and ellipse). On the other hand, the involved singularity issue in dislocation studies is challenging, even intractable for commercial finite element software. Using the numerical equivalent inclusion method (NEIM) in conjunction with the Fast Fourier Transforms technique, this work presents an effective computational scheme for evaluating the interaction energy between an edge dislocation and inhomogeneities. The proposed computational method may successfully circumvent the numerical singularity. The results of norm analyses on relative errors demonstrate that the stress disturbance field caused by the impurity has a great influence on the final solutions, especially when the dislocation is located in the neighborhood of the inhomogeneity. The proposed method in this work shows excellent numerical convergence and stability, and appears to be convenient and efficient for handling arbitrarily shaped inhomogeneities interacting with an edge dislocation.
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图 1 任意形状杂质与刃型位错的相互作用示意图
Figure 1. Schematic of an arbitrarily shaped inhomogeneity interacting with an edge dislocation
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图 2 杂质-位错问题中的Eshelby等效夹杂法
Figure 2. Schematic of Eshelby's equivalent inclusion method for the interaction between an edge dislocation and an inhomogeneity
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图 3 杂质-位错问题中的交互能和位错受力的计算流程图
Figure 3. Flow-chart of the present computational scheme for solving the interaction energy and force on dislocation due to an edge dislocation interacting with an inhomogeneity
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图 4 圆形杂质与刃型位错的相互作用示意图
Figure 4. Schematic of a circular inhomogeneity interacting with an edge dislocation
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图 5 无量纲交互能的变化图，其中位错位置
ξ/a 在x1轴上从1.1移动到1.6Figure 5. Variation of the normalized interaction energy with dislocation position varying along x1-axis from 1.1 to 1.6
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图 6 无量纲位错受力的变化图，其中位错位置
ξ/a 在x1轴上从1.1移动到1.6Figure 6. Variation of the normalized force with dislocation position varying along x1-axis from 1.1 to 1.6
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图 7 刃型位错与Al2O3交互能的变化图，其中位错位置
ξ/a 在x1轴上从2.1移动到2.6Figure 7. Variation of the interaction energy between edge dislocation and Al2O3 where dislocation position varies along x1-axis from 2.1 to 2.6
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图 8 位错受力的变化图，其中位错位置
ξ/a 在x1轴上从2.1移动到2.6Figure 8. Variation of the force on edge dislocation with dislocation position
ξ/a varying along x1-axis from 2.1 to 2.6![]()
图 9 椭圆形SiC杂质与刃型位错的交互能
Figure 9. Schematic of an elliptical SiC inhomogeneity interacting with an edge dislocation
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图 10 复杂形状杂质与刃型位错的交互能
Figure 10. Schematic of the interaction energy between an inhomogeneity with complex boundary and an edge dislocation
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图 11 复杂形状杂质与刃型位错作用下的位错受力
Figure 11. Schematic of the force due to an inhomogeneity with complex boundary and an edge dislocation
表 1 夹杂物和相应基体的计算参数
Table 1 Computational parameters of an inhomogeneity and the corresponding matrix
参数 基体 SiC Ti-6Al-V 剪切模量μ/GPa 80.8 176.70 41.00 泊松比 ν 0.3 0.16 0.34 
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表 2 本文数值计算方案和零次迭代解之间的相对误差结果对比
Table 2 Comparisons on relative error of results between the present numerical scheme and 0th solution
数值方法 材料 相互作用能/(%) 位错受力/(%) δ∞(ΔW) δ2(ΔW) δ∞(F1) δ2(F1) 本文解 SiC 0.7604 0.9643 8.7324 6.8297 Ti-6Al-V 1.0585 1.1993 9.3380 5.9627 Al2O3 1.0452 1.3100 6.4230 3.3556 零次迭代解 SiC 39.4298 38.7584 27.5083 32.3126 Ti-6Al-V 20.1655 19.4616 29.7823 26.1940 Al2O3 20.5311 18.5288 28.4090 24.4514
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