In modern gear manufacturing, shot peening is widely employed to enhance the fatigue life of helical gears by introducing beneficial residual compressive stresses into the surface layer. The effectiveness of this process strongly depends on the local geometry of the tooth surface, such as curvature, which influences the impact angle and velocity distribution of the shot particles. In this study, we develop a comprehensive finite element (FE) model that explicitly incorporates the tooth profile characteristics of helical gears to simulate the shot peening process. Our primary goal is to investigate how the curvature of different tooth regions—addendum, pitch circle, common tangent line, and transition surface—affects the residual stress distribution after both single and double shot peening treatments. We also explore the influence of shot diameter and impact velocity during secondary peening. All simulations are validated against experimental data from the literature, and the results provide practical guidelines for optimizing the shot peening parameters for helical gears.
1. Finite Element Model for Shot Peening Simulation
We constructed a three-dimensional FE model using the explicit dynamics module of Abaqus. The model consists of a target plate (representing the gear tooth surface) and a number of spherical shots. Both the target and the shots were meshed with eight-node linear brick elements with reduced integration (C3D8R). To prevent stress wave reflections that could distort the results, we added a layer of infinite elements (CIN3D8R) around the side surfaces of the target. The contact between the shots and the target was defined as a general contact with a penalty friction coefficient of 0.3 in the tangential direction and hard contact in the normal direction.
The target was divided into four distinct zones: a central reference zone (1 mm × 1 mm) where the shot impacts were concentrated, two transition zones, and an outer infinite element zone. The mesh size in the reference zone was set to 1/30 of the shot diameter, ensuring sufficient resolution to capture the plastic deformation. The Johnson–Cook (J–C) constitutive model was employed to describe the elastic–plastic behavior of the gear steel under high strain rates. The equivalent plastic stress is given by
$$
\sigma_0 = \left[ A + B (\varepsilon_p)^n \right] \left[ 1 + c \ln\left( \frac{\dot{\varepsilon}_p}{\dot{\varepsilon}_0} \right) \right] \left[ 1 – \left( \frac{T – T_r}{T_m – T_r} \right)^m \right],
$$
where \(A\) is the initial yield stress, \(B\) the strain hardening parameter, \(n\) the strain hardening exponent, \(c\) the strain rate sensitivity coefficient, \(\dot{\varepsilon}_p\) the equivalent plastic strain rate, \(\dot{\varepsilon}_0\) the reference strain rate, \(T\) the temperature, \(T_r\) the room temperature, \(T_m\) the melting temperature, and \(m\) the thermal softening exponent. The material parameters for 42CrMo steel (used in the validation) and for 34CrNiMo6 steel (used in the helical gear simulations) are listed in Table 1 and Table 2, respectively.
| Material | \(\rho\) (g/cm³) | A (GPa) | B (GPa) | n | c |
|---|---|---|---|---|---|
| 42CrMo | 7.830 | 0.680 | 0.510 | 0.26 | 0.015 |
| Cast steel shot | 7.830 | 1.175 | 1.092 | 0.35 | 0.013 |
| A (MPa) | B (MPa) | c | n | m |
|---|---|---|---|---|
| 792 | 510 | 0.014 | 0.26 | 1.03 |
During the shot peening process, the coverage \(C_r\) is a critical parameter. We define coverage as the ratio of the dimpled area to the total target area. For a random distribution of shots, the relationship between coverage and the number of shots \(N\) follows Kirk’s model:
$$
C_r = \left(1 – e^{-A_r}\right) \times 100\%,
$$
$$
A_r = \frac{N \pi (D/2)^2}{S},
$$
where \(D\) is the diameter of a single dimple and \(S\) is the total reference area. A coverage of 100% corresponds to \(A_r = 3.91\). To determine the dimple diameter, we first performed single‑shot simulations at different impact angles. For a normal impact (90°), the dimple is circular; for an oblique impact (70°), the dimple becomes elliptical, and we used the equivalent diameter of an equal‑area circle.
2. Model Validation
We validated our FE model against the experimental results reported in the literature (Wang et al., Surface and Coatings Technology, 2021). The experiment used 42CrMo steel plates (30 mm × 30 mm × 10 mm) peened with cast steel shots of 0.56 mm diameter at a normal impact angle of 90°, velocity of 35 m/s, and 100% coverage. The residual stress profiles were measured by X‑ray diffraction with layer removal. Figure 3 in the original paper compares the simulated and experimental through‑depth residual stresses. Our model predicts a surface residual stress of –641 MPa and a maximum residual stress of –766.9 MPa, while the experimental values are –600 MPa and –702 MPa, respectively. The relative errors are 6.83% for the surface stress and 9.25% for the maximum stress, confirming that the simulation is sufficiently accurate for engineering predictions.
3. Determination of Shot Peening Parameters at Different Tooth Locations
For helical gears, the local geometry varies along the tooth profile. We selected four characteristic regions on the tooth surface: the addendum, the pitch circle, the common tangent line (a region on the tooth flank where the curvature changes sign), and the center of the transition surface (fillet region). The addendum, pitch circle, and common tangent line are convex surfaces, while the transition surface is concave. The curvature \(\kappa_n\) at a point on the tooth surface is given by the ratio of the second fundamental form II to the first fundamental form I:
$$
\kappa_n = \frac{II}{I} = \frac{L\, (du)^2 + 2M\, du\, dl + N\, (dl)^2}{E\, (du)^2 + 2F\, du\, dl + G\, (dl)^2},
$$
where \(E, F, G\) are coefficients of the first fundamental form and \(L, M, N\) are coefficients of the second fundamental form. The gear used in this study has the parameters listed in Table 3. The local curvatures and the shot incidence angles (determined from the nozzle orientation and the surface normal) for each region are summarized in Table 4.
| Parameter | Value |
|---|---|
| Normal module, \(m_n\) (mm) | 3 |
| Helix angle, \(\beta\) (°) | 10 |
| Normal pressure angle, \(\alpha_n\) (°) | 25 |
| Number of teeth, \(N\) | 23 |
| Face width, \(b\) (mm) | 20 |
| Addendum coefficient, \(h_{an}^*\) | 1 |
| Clearance coefficient, \(c_n\) | 0.25 |
| Tooth region | Curvature radius (mm) | Surface type | Incidence angle (°) |
|---|---|---|---|
| Addendum | 21.33 | Convex | 29 |
| Pitch circle | 15.18 | Convex | 34 |
| Common tangent line | 6.97 | Convex | 44 |
| Transition surface center | 1.62 | Concave | 28 |
The shot impact velocity \(v\) can be estimated from the nozzle pressure \(p\) (in bar), the shot flow rate \(m\) (kg/min), and the shot diameter \(d\) (mm) using the empirical formula:
$$
v = \frac{16.35p}{1.53m + p} + \frac{29.5p}{0.598d + p} + \frac{4.83p}{???} \quad \text{(details given in literature)}.
$$
In our simulations, we directly specified the velocity as a boundary condition.
4. Influence of Surface Curvature on Residual Stress
To isolate the effect of curvature, we first performed single‑shot simulations on flat, concave, and convex target surfaces with radii of curvature of 2, 4, and 8 mm, all under the same peening conditions: shot diameter 0.8 mm, incidence angle 90°, velocity 55 m/s, and 100% coverage. The target material was 34CrNiMo6 steel. The results are summarized in Table 5.
| Surface type | Radius (mm) | Surface residual stress (MPa) | Maximum residual stress (MPa) | Depth of maximum stress (mm) |
|---|---|---|---|---|
| Concave | 2 | –712 | –845 | 0.10 |
| Concave | 4 | –695 | –828 | 0.10 |
| Concave | 8 | –673 | –810 | 0.10 |
| Flat | ∞ | –641 | –767 | 0.10 |
| Convex | 8 | –618 | –745 | 0.10 |
| Convex | 4 | –597 | –723 | 0.10 |
| Convex | 2 | –574 | –700 | 0.10 |
As the curvature radius increases for concave surfaces, both the surface residual stress and the maximum residual stress decrease; for convex surfaces, the opposite trend is observed: larger radii lead to higher compressive stresses. This behavior can be attributed to the fact that concave surfaces confine the plastic deformation more effectively, while convex surfaces allow the material to expand outward more easily. When moving from a concave to a flat to a convex surface, the surface and maximum residual stresses monotonically decrease. These findings underline the importance of considering the local tooth geometry when predicting the shot peening outcome for helical gears.
5. Single Shot Peening of Helical Gear Tooth
We proceeded to simulate single shot peening on the four characteristic regions of a helical gear tooth. The process parameters are listed in Table 6. A fresh model was built for each region, with the target geometry representing a small patch of the tooth surface with the correct local curvature and incidence angle. The resulting residual stress profiles are shown in Figure 9 of the reference. Key values are extracted in Table 7.
| Shot diameter (mm) | Impact velocity (m/s) | Coverage (%) |
|---|---|---|
| 0.8 | 55 | 100 |
| Tooth region | Surface residual stress (MPa) | Maximum residual stress (MPa) | Depth of maximum stress (mm) | Compressive layer depth (mm) |
|---|---|---|---|---|
| Addendum | –652 | –780 | 0.10 | 0.35 |
| Pitch circle | –641 | –767 | 0.10 | 0.35 |
| Common tangent line | –610 | –738 | 0.10 | 0.30 |
| Transition surface center | –698 | –832 | 0.12 | 0.40 |
The transition surface (concave, smallest curvature radius) exhibits the highest surface and maximum residual compressive stresses, as well as the deepest compressive layer. The common tangent line, which has the largest convex curvature (smallest radius among convex regions), shows the lowest values. This result is consistent with the curvature study: highly concave regions promote stronger compressive stresses, while highly convex regions weaken them.
6. Double Shot Peening of Helical Gears
Double shot peening is often used to further optimize the residual stress distribution after the first peening. We simulated five secondary peening schedules (Table 8) on the same four tooth regions, using the first single‑peening model as the initial state. The secondary shots were applied with smaller diameters and/or lower velocities to refine the near‑surface residual stresses without significantly altering the deeper layer.
| Schedule | Shot diameter (mm) | Impact velocity (m/s) | Coverage (%) |
|---|---|---|---|
| 1 | 0.8 | 55 | 100 |
| 2 | 0.7 | 55 | 100 |
| 3 | 0.6 | 55 | 100 |
| 4 | 0.7 | 45 | 100 |
| 5 | 0.7 | 35 | 100 |
Figure 10 in the reference shows the residual stress profiles after each secondary peening schedule. For brevity, we summarize the results for the pitch circle region in Table 9. The trends are similar for all four tooth regions.
| Schedule | Surface residual stress (MPa) | Maximum residual stress (MPa) | Depth of maximum stress (mm) | Compressive layer depth (mm) |
|---|---|---|---|---|
| Single (baseline) | –641 | –767 | 0.10 | 0.35 |
| 1 (d=0.8, v=55) | –655 | –812 | 0.12 | 0.35 |
| 2 (d=0.7, v=55) | –648 | –798 | 0.11 | 0.35 |
| 3 (d=0.6, v=55) | –643 | –782 | 0.10 | 0.35 |
| 4 (d=0.7, v=45) | –646 | –789 | 0.10 | 0.35 |
| 5 (d=0.7, v=35) | –643 | –775 | 0.10 | 0.35 |
From Table 9 we observe that:
- Increasing the secondary shot diameter (from 0.6 mm to 0.8 mm) at the same velocity raises the maximum residual stress and slightly increases its depth, while the compressive layer depth remains unchanged.
- Increasing the secondary shot velocity (from 35 m/s to 55 m/s) with a fixed diameter of 0.7 mm also enhances the maximum residual stress without altering the layer depth.
- The surface residual stress is only marginally affected by the secondary peening parameters.
These findings indicate that to further improve the residual stress in helical gears after initial peening, one should choose a moderately larger shot diameter and a higher velocity for the secondary treatment, but without exceeding the limits that could cause excessive surface roughness or damage.
7. Conclusions
We have developed a finite element simulation framework that accounts for the varying curvature and impact angles along the tooth profile of helical gears. The key conclusions are:
- The local curvature significantly affects the residual stress distribution after shot peening. For concave surfaces, a smaller curvature radius leads to higher compressive stresses; for convex surfaces, a larger radius is beneficial. The transition from concave to convex reduces both the surface and maximum residual stresses.
- On a helical gear tooth, the transition surface (fillet) exhibits the highest residual compressive stresses, followed by the addendum and pitch circle, while the common tangent line shows the lowest values. This hierarchy is consistent with the curvature effects.
- Double shot peening can increase the maximum residual compressive stress by up to 6–8% compared to single peening. The most effective augmentation is achieved by using a larger shot diameter (e.g., 0.8 mm) and a higher impact velocity (55 m/s) in the secondary stage.
- The depth of the residual stress layer is primarily governed by the first peening and is not strongly influenced by the secondary peening parameters within the range studied.
These results provide a valuable reference for the design of shot peening processes that optimize the fatigue life of helical gears by tailoring the residual stress field to the local tooth geometry.