Shot peening earns its place in the routing of aerospace spiral bevel gears because the plastic deformation it imposes generates a compressive residual stress field that suppresses fatigue crack initiation in a part running at high speed under heavy load. That same deformation, however, reworks a flank that has already been ground or lapped to a controlled finish. Push the intensity too far and the surface roughness climbs steeply, the lubricating film thins locally, and part of the contact-fatigue benefit that justified the operation is spent on a rougher surface. Excessive peening can additionally introduce surface microcracks. The usable process window is therefore narrow, and it is normally located by peening coupons or actual gears, sectioning them, and measuring the result — a loop that consumes hardware and time and says little about why the roughness moved.
What the process planner needs instead is a way to compute the as-peened topography directly on the real tooth flank, so that coverage, shot size and velocity can be ranked before any metal is cut. 
Why curved tooth flanks break the standard roughness model
Most numerical work on peening-induced roughness has been built on random multi-shot finite element models applied to flat plates or cylindrical surfaces. In such open geometry the probability that one shot strikes another is small, so shot–shot collision is dropped from the model and the impact stream is represented by randomly distributed indenters with a prescribed velocity. Those assumptions do not survive the transition to a gear. A spiral bevel flank is a curved, twisted surface, and the gap between neighbouring teeth forms a narrow channel. Shot rebounding from one flank crosses the incoming stream, colliding with it, losing energy and being deflected; the number, angle and speed of impacts per unit area then differ markedly between the heel, the mid-face and the toe. Grazing impacts that barely deform the surface also become common.
The practical consequence is that the impact statistics must be computed rather than assumed. A discrete element method (DEM) stage resolves the flight of shot from the nozzle to the flank and records where and how fast each particle arrives; a finite element (FE) stage then reproduces the indentation those arrivals produce in the actual gear steel. Coupling the two sequentially gives a chain from machine settings to surface topography, with the DEM output acting as the loading for the FE model. This is the approach developed here for a production-type spiral bevel gear, validated against interferometric measurements, and then used to map the influence of peening time, shot velocity and shot diameter.
Specimen, peening conditions and surface measurement
The test gear was a spiral bevel gear in AISI 9310 high-strength alloy steel, carburized and quenched to a tooth surface hardness of 62 HRC. Its composition is given in Table 1. Peening was carried out on a compressed-air machine with S110 cast steel shot of 0.3 mm nominal diameter and 55–62 HRC hardness, at an Almen intensity of 0.196 A. Shot flow was 9 kg/min at an air pressure of 4.5 bar (0.45 MPa). The nozzle traversed at 150 mm/min along the face width, its axis perpendicular to the tooth root and 180 mm from it, while the gear rotated with the worktable at 10 r/min.
| Element | C | Si | Mn | P | S | Cr | Ni |
|---|---|---|---|---|---|---|---|
| Content (wt %) | 0.12 | 0.25 | 0.55 | 0.003 | 0.002 | 1.25 | 3.34 |
After peening, the gear was sectioned by wire electrical discharge machining and ultrasonically cleaned. Topography was measured on a Wyko NT9100 white-light interferometer with a 20× objective. Measurement windows of 0.5 mm along the face width by 0.5 mm along the tooth height were placed at three positions — near the heel, at mid-face and near the toe — with three windows at each position. The three windows on the convex flank, designated a, b and c, provided the comparison used for validation. The three-dimensional arithmetic mean height Sa of the unpeened flanks was 0.4 μm.
A two-stage DEM–FEM workflow
Shot transport. The discrete element model, built in EDEM, represents the nozzle and the toothed flank, the latter imported as a surface mesh generated in Hypermesh so that each impact can be tagged to a specific flank element. The nozzle sits 180 mm from the flank with its axis normal to the root and travels along the face width. The initial shot velocity follows the empirical relation
$$v = \frac{16.35\,P^{1.53}}{q_m + P} + \frac{29.5\,P^{0.598}}{d + P} + 4.83\,P$$
where d is shot diameter in mm, qm is shot flow in kg/min and P is nozzle pressure in bar. Pressure dominates the three variables, so raising it is the most direct way to raise impact velocity and hence intensity. Applied to the test conditions above, the relation returns a velocity of roughly 50 m/s, consistent with the 30–50 m/s range explored in the parameter study.
From machine frame to flank frame. The DEM solver records impact positions and velocities in the machine tool frame Sm, which rotates with the worktable, whereas the FE model requires them in the gear frame Sg. With the gear turning at angular velocity ω, the rotation accumulated in time t is α = ωt and the transformation between the two frames is
$$\mathbf{M}_{mg} = \begin{bmatrix} \cos\alpha & \sin\alpha & 0 \\ -\sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
A local frame Sf is then constructed on the flank from three adjacent mesh nodes O, P and Q, taking Xf along the edge OP and Zf along the surface normal, so that the normal velocity component of each impact can be evaluated. Impacts whose normal velocity is very low contribute little to the indentation and are filtered out; only the retained events, with their positions, velocities and per-unit-area counts, are passed to the FE stage.
Indentation. The FE model in Abaqus uses a target block of 1 mm × 1 mm × 0.5 mm meshed with C3D8R elements. Two central regions carrying the impacts are refined to 10 μm × 10 μm × 10 μm, a graded transition region follows, and the block is surrounded by CIN3D8 infinite elements that absorb the outgoing stress waves and prevent reflections from the artificial boundaries. The base of the block is fully fixed. Shot and target interact through surface-to-surface contact with hard normal behaviour and penalty friction of 0.15, a value whose variation between 0.1 and 0.5 has little effect on the resulting plastic strain. Shot of 0.3 mm diameter is modelled as an elastic–plastic body with isotropic hardening and a yield strength of 1 400 MPa. Both materials use E = 210 GPa, ν = 0.3 and ρ = 7 800 kg/m³. The gear steel follows the Johnson–Cook law
$$\sigma = \left(A + B\varepsilon^{n}\right)\left(1 + C\ln\frac{\dot{\varepsilon}}{\dot{\varepsilon}_{0}}\right)\left[1 – \left(T^{*}\right)^{m}\right], \qquad T^{*} = \frac{T – T_r}{T_q – T_r}$$
with A = 1 234.38 MPa, B = 881 MPa, C = 0.018, n = 0.238 and m = 0.686, a reference temperature of 25 °C and a melting point of 1 510 °C. The measured pre-peening roughness is introduced as a predefined field on the target surface, so the simulation starts from the real flank rather than an ideal plane. Rayleigh damping is applied to suppress residual stress-wave oscillation. Impact centres are placed within the refined region by a random-number routine in Python, reproducing the stochastic spacing of real impacts.
Surface roughness is evaluated from the deformed nodal heights as
$$S_a = \frac{1}{n}\sum_{i=1}^{n}\left|Z_i\right|$$
where Zi is the height of node i and n is the number of nodes in the evaluation window.
Validation against measured topography
Running the model with the experimental process settings reproduces the character of the measured surfaces: irregular fields of overlapping craters whose depth and spacing vary from window to window. The quantitative comparison is collected in Table 2. Measured Sa on the convex flank ranged from 0.741 to 0.938 μm, while the simulated values ranged from 0.883 to 1.09 μm, giving deviations of 13.5% to 19.2% and confirming that the coupled workflow captures the process at engineering accuracy.
| Region | Measured Sa (μm) | Simulated Sa (μm) | Deviation (%) |
|---|---|---|---|
| a (heel side) | 0.938 | 1.09 | 16.2 |
| b (mid-face) | 0.817 | 0.927 | 13.5 |
| c (toe side) | 0.741 | 0.883 | 19.2 |
The deviation is one-sided: the model overpredicts in all three windows, so what is demonstrated is a bounded bias rather than random scatter. Part of that bias is plausibly traceable to the idealization of the media as identical spheres of a single diameter, whereas S110 shot is a size distribution with fines and occasional broken particles that would blunt the deepest craters. Region b, at mid-face on the convex flank, carries the main contact zone and was therefore adopted as the reference window for the parameter study that follows.
Coverage saturation and the non-monotonic roughness curve
Coverage was computed from the DEM impact statistics as the fraction of flank area occupied by indentations. At an initial shot velocity of 40 m/s, the surface reached 98% coverage — treated as full coverage — after 50 s of peening, and 200% coverage after 100 s. Roughness did not follow coverage monotonically. Starting from 0.4 μm, Sa rose to 1.01 μm at full coverage and then fell slowly to 0.97 μm at 200% coverage. The two regimes have different physics: while fresh material is still being struck, every new impact creates a new crater on a previously undisturbed surface, adding height variance; once the surface is saturated, incoming shot lands on existing craters, flattening the rims and redistributing ridge material rather than creating new relief. Diminishing the roughness by extending peening time is thus possible, but only marginally and at the cost of additional cold work and machine time.
| Condition (40 m/s, 0.3 mm shot) | Coverage | Sa (μm) |
|---|---|---|
| Before peening | — | 0.40 |
| 50 s | ≈98% (full) | 1.01 |
| 100 s | ≈200% | 0.97 |
Shot velocity
Holding the other settings fixed, peak Sa increased with impact velocity: from 0.90 μm at 30 m/s to 1.01 μm at 40 m/s and 1.10 μm at 50 m/s. Kinetic energy scales with the square of velocity, so a faster particle drives a larger plastic zone and a deeper, wider crater at the same shot mass. Because nozzle pressure is the dominant term in the velocity relation, any adjustment made to reach a required Almen intensity by raising pressure will also raise flank roughness — the two responses are coupled and cannot be specified independently.
Shot diameter
Diameter proved the stronger of the two variables. At 50 m/s, 0.6 mm shot drove Sa to a peak of 1.45 μm at about 7 s before settling near 1.23 μm, while 0.9 mm shot reached 2.00 μm within about 2 s and stabilized near 1.65 μm. Mass scales with the cube of diameter, so the kinetic energy delivered per impact grows steeply as the media coarsens; the indentation radius expands accordingly and the resulting relief is coarser. The larger shot also saturates the surface far sooner, because each indentation covers a greater area at the same impact count rate, which compresses the rising branch of the roughness curve into the first few seconds of the cycle.
| Variable | Setting | Peak Sa (μm) | Time to peak | Sa after prolonged peening (μm) |
|---|---|---|---|---|
| Shot velocity | 30 m/s | 0.90 | — | — |
| 40 m/s | 1.01 | ≈50 s | 0.97 at 100 s | |
| 50 m/s | 1.10 | — | — | |
| Shot diameter (50 m/s) | 0.6 mm | 1.45 | ≈7 s | ≈1.23 |
| 0.9 mm | 2.00 | ≈2 s | ≈1.65 |
Implications for process design
Two conclusions follow directly from the ranking of the variables. First, media size should be treated as the primary roughness lever and specified as the smallest diameter that still delivers the required intensity; moving from 0.3 mm to 0.9 mm shot more than doubled the stabilized Sa, a penalty no amount of additional peening time recovers. Second, coverage should be controlled as a target rather than a lower bound. Peening well beyond full coverage buys a slow, small reduction in Sa while accumulating cold work, and the simulations show the surface is already at its roughest around the moment coverage is completed — the least favourable point in the cycle coincides with the condition most often specified in practice.
The same simulations also show that peening narrows the spread of surface finish. Every peened condition gave Sa above the 0.4 μm pre-peened value, yet the window-to-window scatter was smaller after peening than before, since a stochastic crater field overwrites the deterministic marks left by machining. Where flank-to-flank consistency matters more than absolute smoothness, that homogenization is a real benefit of the process, not merely a side effect.
Scope of the prediction
The validated workflow is an engineering tool with bounded accuracy: deviations of 13.5%–19.2% are adequate for ranking parameters and screening process windows, but not for certifying a finish specification without experimental confirmation. The description is also restricted to the amplitude parameter Sa and to a single gear geometry and steel; hybrid and functional roughness parameters, residual stress evolution, and the effect of a distributed shot size have not been folded into the chain. Within those limits, the DEM–FEM sequence supplies what trial peening cannot — a per-impact explanation of why a given set of machine settings produces the topography it does, on a curved flank where the shot stream interacts with itself inside the tooth gap.