FULL SOLUTION FOR CHARACTERIZING STRESS FIELDS NEAR THE TIP OF MODE-I CRACK UNDER PLANE AND POWER-LAW PLASTIC CONDITIONS
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摘要: 在航空航天、船舶、石油管道和核电等领域, 服役结构或部件在长期极端条件下运行, 不可避免地会产生裂纹, 因此, 为研究含裂纹结构的准静态断裂行为, 必须了解裂纹尖端附近区域的应力应变场特点. 对于幂律材料裂纹构元, 研究平面应变和平面应力条件下I型裂纹尖端应力场的解析分布. 基于能量密度等效和量纲分析, 推导了能量密度中值点代表性体积单元 (representative volume element, RVE)的等效应力解析方程, 并定义其为应力因子, 进而针对有限平面应变和平面应力紧凑拉伸(compact tension, CT)试样和单边裂纹弯曲(single edge bend, SEB)试样, 以应力因子作为应力特征量, 并构造用于表征裂尖等效应力等值线的蝶翅轮廓式和扇贝轮廓式三角特殊函数, 提出描述幂律塑性条件下平面I型裂纹尖端应力场的半解析模型. 该半解析模型形式简单, 对CT和SEB试样的裂尖应力场的预测结果与有限元分析的结果比较表明, 两者之间均密切吻合, 模型公式可直接用于预测I型裂纹尖端应力分布, 方便于断裂安全评价和理论发展.Abstract: In the fields of aerospace, ships, oil pipelines and nuclear power, there will be cracks inevitably in structure or component part when running for a long time under extreme conditions. Therefore, it is necessary to explore the features of the stress-strain fields near the crack tip, to study the quasi-static fracture behavior of cracked structures. In this paper, the stress distributions near the tip of mode-I cracked specimens under plane strain and plane stress conditions are studied for power-law hardening material. Based on the energy density equivalence and dimensional analysis, the analytical equation of equivalent stress of representative volume element (RVE) with the median energy density of a finite-dimensions specimen is proposed, and it is defined as the stress factor. Furthermore, for compact tension (CT) and single edge bend (SEB) finite size specimens under plane strain and plane stress conditions, the stress factor is used as a characteristic variable, and a special trigonometric function is assumed to characterize butterfly-wings type or scallop type contour lines of the equivalent stress near the mode-I crack tip, and then a semi-analytical model for compact tension specimens and single edge bend specimens under plane strain and plane stress and fully plastic conditions is proposed to describe the stress fields near the crack tip. As shown in comparing results given by finite element analysis to those predicted by the model for stress fields near the crack tip of the two cracked specimens, all agree well with each other. The semi-analytical model of stress field near the crack tip proposed in this paper is simple in form and accurate in result. It can be directly used to predict the stress distribution near the tip of mode-I crack, which is convenient for fracture safety evaluation and theoretical development.
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图 4 平面应变条件下裂尖附近网格密度对FEA结果的影响
Figure 4. Effects of element meshing size around the crack tip on the FEA results under plane strain conditions
图 5 平面应力条件下裂尖附近网格密度对FEA结果的影响
Figure 5. Effects of element meshing size around the crack tip on the FEA results under plane stress conditions
图 8 载荷P和应力硬化系数K对λsub的影响
Figure 8. Effects of loading P and strain hardening coefficient K on the λsub
图 11 CT试样平面应变条件下裂尖无量纲应力径向分布对比
Figure 11. Comparison with FEA and predicted results by Eq. (25) for normalized stress radial distributions near the crack tip of CT specimens under plane strain conditions
图 12 CT试样平面应力条件下裂尖无量纲应力径向分布对比
Figure 12. Comparison with FEA and predicted results by Eq. (25) for normalized stress radial distributions near the crack tip of CT specimens under plane stress conditions
图 13 SEB试样平面应变条件下裂尖无量纲应力径向分布对比
Figure 13. Comparison with FEA and predicted results by Eq. (25) for normalized stress radial distributions near the crack tip of SEB specimens under plane strain conditions
图 14 SEB试样平面应力条件下裂尖无量纲应力径向分布对比
Figure 14. Comparison with FEA and predicted results by Eq. (25) for normalized stress radial distributions near the crack tip of SEB specimens under plane stress conditions
图 15 CT试样平面应变条件下裂尖无量纲应力环向分布对比
Figure 15. Comparison with FEA and predicted results by Eq. (25) for normalized stress angular distributions near the crack tip of CT specimens under plane strain conditions
图 16 CT试样平面应力条件下裂尖无量纲应力环向分布对比
Figure 16. Comparison with FEA and predicted results by Eq. (25) for normalized stress angular distributions near the crack tip of CT specimens under plane stress conditions
图 17 SEB试样平面应变条件下裂尖无量纲应力环向分布对比
Figure 17. Comparison with FEA and predicted results by Eq. (25) for normalized stress angular distributions near the crack tip of SEB specimens under plane strain conditions
图 18 SEB试样平面应力条件下裂尖无量纲应力环向分布对比
Figure 18. Comparison with FEA and predicted results by Eq. (25) for normalized stress angular distributions near the crack tip of SEB specimens under plane stress conditions
图 19 CT试样平面应变条件下等效应力σeq等效应力等值线对比
Figure 19. Comparison with FEA results predicted by Eq. (26) for contour lines of the equivalent stress σeq near the crack tip of CT specimen under plane strain conditions
图 20 SEB试样平面应变条件下等效应力σeq等效应力等值线对比
Figure 20. Comparison with FEA results predicted by Eq. (26) for contour lines of the equivalent stress σeq near the crack tip of SEB specimen under plane strain conditions
图 21 CT试样平面应力条件下等效应力σeq等效应力等值线对比
Figure 21. Comparison with FEA results predicted by Eq. (26) for contour lines of the equivalent stress σeq near the crack tip of CT specimen under plane stress conditions
图 22 SEB试样平面应力条件下等效应力σeq等效应力等值线对比
Figure 22. Comparison with FEA results predicted by Eq. (26) for contour lines of the equivalent stress σeq near the crack tip of SEB specimen under plane stress conditions
表 1 方程(15)模型参数
Table 1. Parameters of Eq. (15)
Specimen Condition m k5 a/W CT plane strain 2.2800 1.4308 [0.45,0.75] plane stress 2.2800 2.0971 SEB plane strain 1.9560 2.3400 [0.40,0.75] plane stress 1.9560 3.3024 
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表 2 方程(19)参数
Table 2. Parameters of Eq. (19)
Specimen Condition Stress Sub msub0 msub1 a/W CT plane strain σsub eq 0.0000 0.0000 [0.45,0.75] 11 0.0873 1.1981 22 0.0873 1.1981 12 0.0172 0.4345 plane stress eq 0.0000 1.2658 11 0.0000 1.2658 22 0.0000 1.2658 12 0.0000 1.2658 SEB plane strain σsub eq 0.0722 0.5419 [0.40,0.75] 11 −0.244 1.2000 22 −0.0754 0.7258 12 0.0000 0.0000 plane stress eq 0.0000 0.0000 11 0.0000 0.0000 22 0.0000 0.0000 12 −0.0150 −0.2903
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表 3 方程(23)模型参数
Table 3. Parameters of Eq. (23)
Specimen Condition Stress Sub dsub0 dsub1 dsub2 dsub3 a/W CT plane strain σsub eq 1.5013 −0.6940 1.6796 0.0000 [0.45,0.75] 11 1.1851 0.3635 2.9854 3.7610 22 1.2000 0.0000 0.0000 0.0000 12 4.8305 −4.0000 −0.3490 1.8316 plane stress eq 1.1448 0.2524 1.5459 −1.8604 11 1.0655 0.6019 −0.8832 −1.9732 22 −6.0795 −7.0000 −0.4000 4.7810 12 1.4504 −0.04557 −3.0191 −4.0356 SEB plane strain σsub eq 0.9250 0.2067 3.2036 1.3810 [0.40,0.75] 11 1.5339 0.5333 −2.2470 −4.0212 22 8.1002 −7.0000 0.2605 −4.7292 12 0.8052 0.1382 2.7227 1.4797 plane stress eq 1.0863 0.2285 −1.8062 −1.0613 11 0.9674 0.4968 2.6107 4.5223 22 −3.0386 3.9167 0.4285 1.5112 12 1.1793 0.0306 4.9568 −0.1173
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表 4 模型参数
Table 4. Parameters of model
Specimen Condi
tionStress Sub qsub0 qsub1 qsub2 qsub3 qsub4 csub0 csub1 bsub0 bsub1 a/W CT plane strain σsub eq 1.00 − − − − 2.1842 0.2756 −0.4266 0.2017 [0.45,0.75] 11 π/2 0.3333 0.3333 0.9999 π/2 2.3397 1.5564 0.1430 −0.4930 22 π/2 0.6667 0.6667 2.0001 π/2 0.5754 6.7272 −0.1684 −0.0309 12 0.00 1.0000 2.0000 0.0000 π/2 0.2153 4.9742 0.0698 −0.1902 plane stress eq 0.8500 − − − − 0.6707 1.3870 −0.3550 −0.2910 11 π/2 0.5550 0.5550 1.6550 π/2 −0.1669 −9.5370 −0.53 0.0000 22 π/2 0.8150 0.8150 2.4450 π/2 0.1860 8.9576 −0.3200 0.0000 12 0.00 1.0000 2.0000 0.0000 π/2 0.2261 5.3946 −0.1803 0.0000 SEB plane strain σsub eq 1.00 − − − − 1.9865 0.2955 −0.3812 0.1566 [0.45,0.75] 11 π/2 0.3561 0.3561 1.0683 π/2 1.8385 0.5354 0.0117 −0.2631 22 π/2 0.7683 0.7683 2.3049 π/2 0.2759 3.9625 −0.2346 0.1882 12 0.00 1.0000 2.0000 0.0000 π/2 0.1182 3.2612 0.1091 −0.1175 plane stress eq 0.85 − − − − 0.2461 0.6570 −0.0420 −0.0630 11 π/2 0.6000 0.6000 1.8000 π/2 −0.04845 −8.4969 0.4000 −0.6200 22 π/2 0.8033 0.8033 2.4099 π/2 0.0030 146.6888 1.5301 −1.5831 12 0.00 1.0000 2.0000 0.0000 π/2 0.1377 2.6570 −0.0551 0.0465
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