# Stress intensity factor of helical gear root crack and influence of J-integral simulation position

Three dimensional crack models at different positions are established on the helical gear, and the stress intensity factor and J-integral at the crack tip are calculated by finite element software. The coordinate system is built in the center of the tooth root surface, and the z-axis is along the helical direction of the helical gear. Based on this coordinate system, the positions of – 6 mm, – 4 mm, – 2 mm, 0 mm, 2 mm, 4 mm and 6 mm are moved along the z-axis respectively, and a three-dimensional semi elliptical crack model with length and depth of 1.0 mm is established. After 500 N · m torque is applied to the driving wheel, the stress intensity factor and J-integral are calculated respectively.

The distribution of crack front K1 is shown in Fig. 1 (a). It can be seen from Fig. 1 (a) that the crack front K1 increases with the increase of the distance from the tooth root center to the negative direction. It can be seen that the curve approximately moves upward, while the crack front K1 hardly changes with the increase of the distance from the tooth root center to the positive direction; all the curves maintain the change trend of high middle low on both sides. It can be seen from Fig. 1 (b) that when moving from the root center to the negative direction, with the increase of the distance, the reverse maximum value and positive maximum value of leading edge crack K2 increase; while when moving from the root center to the positive direction, the crack leading edge K2 increases as a whole; when the position moves from – 6 mm to 6 mm, it can be seen that the curve approximately rotates clockwise around a certain point. It can be seen from Fig. 1 (c) that with the increase of the distance from the root center to the negative direction, the leading edge crack K3 decreases in the positive direction and increases in the opposite direction; while with the increase of the distance from the root center to the positive direction, the leading edge crack K3 increases in the positive direction and decreases in the opposite direction; when the position is moved from – 6 mm to 6 mm, it can be seen that the curve is approximately moving upward.

The distribution of J-integral is shown in Figure 2. It can be seen from Figure 2 that with the increase of the distance from the center of the tooth root to the negative direction, the J-integral also increases, while with the increase of the distance from the center of the tooth root to the positive direction, the increase or decrease of the J-integral is not obvious, and there is almost no change; all the curves in Figure 2 keep the change trend of high middle low on both sides. It can be seen from Fig. 1 (a) and Fig. 2 that the variation law of K1 stress intensity factor and J-integral is basically the same.

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