Solving Stress Intensity Factor of a Non-penetrating Dedendum Crack Using Weight Function Method
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摘要: 齿轮齿根是具有非穿透性的三维裂纹，其应力强度因子计算求解难度大且过程复杂。对此应用片条合成权函数法将三维裂纹问题转化为一系列“等效的”薄片单元二维裂纹问题。根据作用在薄片单元上的法向力按照其啮合刚度进行分配，求出作用于每个薄片单元上的法向力，再运用二维裂纹权函数法计算每个薄片单元裂纹应力强度因子。其求解结果与有限元法计算结果十分吻合，最大误差为3.9%。Abstract: A gear tooth root has non-penetrating three-dimensional cracks, its stress intensity factor calculation is difficult and the process is complex. In this study, the three-dimensional crack problem is transformed into a series of "equivalent" two-dimensional crack problems of thin section element by using the weight function method of strip composition. According to the normal force acting on tooth root, the normal force acting on each thin section element was calculated based on its meshing stiffness, and then the crack stress intensity factor of each thin section element was calculated by using the two-dimensional crack weight function method. The calculated results of the "equivalent" solution are in good agreement with those of finite element method, and the maximum error is 3.9%.
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图 1 片条合成权函数法示意图
Figure 1. Schematic diagram of the slice-synthesis weight function method
图 2 齿根薄片单元力学模型
Figure 2. The mechanical model of the slice element of the cracked dedendum
图 4 齿顶法向力作用时裂纹尖端各点的应力强度因子
Figure 4. Stress intensity factors at each crack tip under normal force at gear addendum
图 6 啮合过程中齿根裂纹应力强度因子变化图
Figure 6. Variation of stress intensity factor of crack dedendum in the meshing process
表 1 齿根边缘裂纹权函数的
$ {\beta _i}\left( a \right) $ 系数[12]Table 1. Coefficient of root edge crack weight function [12]
$ a $ $ {\beta _1}\left( a \right) $ $ {\beta _2}\left( a \right) $ $ {\beta _3}\left( a \right) $ $ {\beta _4}\left( a \right) $ 0.01 2.0000 −4.3805 4.7674 −1.6455 0.05 2.0000 −1.5117 4.5540 −1.4116 0.10 2.0000 −0.0074 4.4334 −1.2872 0.20 2.0000 1.8811 2.1992 −0.7144 0.30 2.0000 3.8976 0.9383 −0.3278 0.40 2.0000 5.5887 2.6754 −0.5625 0.50 2.0000 7.8498 4.9653 −0.8697 0.60 2.0000 11.938 7.2380 −1.0518 0.70 2.0000 17.311 15.699 −2.3857 0.80 2.0000 23.436 36.751 −6.1878 0.85 2.0000 30.375 49.189 −8.2129 0.90 2.0000 54.712 60.001 −8.7527 
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表 2 权函数法与有限元法应力强度因子对比
Table 2. Comparison of stress intensity factors between weight function method and finite element method
轴向单元数 权函数法/
（$\rm MPa\cdot {\rm{m}}^{\tfrac{1}{2}} $）有限元法/
（$\rm MPa\cdot {\rm{m}}^{\tfrac{1}{2}} $）相对误差/
%1 3.5695 3.7122 −3.9 2 2.9331 2.9785 −2.6 3 2.2405 2.2107 1.3 4 1.2956 1.2626 2.6 5 0.1037 0.1047 1.1
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