Contact Fatigue, Pitting and Spalling in Planetary Gear Sets

2. Hertzian contact at the gear mesh

2.1 Reduction to equivalent cylinders

At any instant, two involute flanks touch along a line, and in the neighborhood of that line each flank is well approximated by a cylinder whose radius equals the local radius of flank curvature. For spur and helical gearing the pair reduces to a single equivalent cylinder of relative radius R against a rigid plane:

1/R = 1/ρ₁ ± 1/ρ₂ — plus for an external mesh (convex on convex), minus for an internal mesh (convex planet on concave ring).

This single line is where most of the useful engineering lives. The sign change is the reason the ring-gear mesh in a planetary set is rarely the pitting-critical mesh: making ρ₂ negative makes R larger — often several times larger — and contact pressure falls accordingly.

2.2 Contact pressure and how it scales

For line contact the classical Hertz solution gives a semi-elliptical pressure distribution with

where Z_E = √[1/(π·((1−ν₁²)/E₁ + (1−ν₂²)/E₂))] = 189.8 √MPa for steel on steel. Two consequences matter more than the formula itself:

• Pressure scales with the square root of load. Doubling transmitted torque raises flank stress by only ~41 %. This is why flank stress alone is a poor intuition for durability — the steep S–N slope does the rest of the work (§7).
• Pressure scales with the inverse square root of R. Radius of curvature, and with it R, is smallest near the pinion root/start of active profile, so the lower flank of the sun pinion is typically the highest-stressed region of a planetary stage — and also the region with the highest specific sliding.

Contact width is small. For an illustrative case of w′ = 500 N/mm and R = 20 mm, σ_H ≈ 950 MPa and b ≈ 0.34 mm. The entire stressed volume that decides whether the tooth survives is therefore a strip under a millimeter wide and a few millimeters deep.

3. The subsurface stress field

The failure-relevant quantity is not the normal pressure but the shear stress it generates below the surface. For frictionless two-dimensional line contact with ν = 0.3, the standard elasticity solution (Johnson, 1985) gives:

• maximum shear stress τ_max ≈ 0.30 p₀ at a depth z ≈ 0.78 b;
• maximum orthogonal (fully reversing) shear amplitude ≈ 0.25 p₀ at z ≈ 0.50 b, offset laterally by ≈ 0.87 b;
• von Mises equivalent stress peaking at ≈ 0.56 p₀ near z ≈ 0.70 b;
• at the surface directly under the load, τ_max ≈ 0.20 p₀ — a local minimum, not a maximum.

There is longstanding and unresolved debate over which of these is the correct initiation criterion. The orthogonal shear amplitude is favored in the rolling-bearing tradition because it reverses sign as the contact passes, giving a true alternating stress; the von Mises or maximum-shear criterion is favored in gear work. Both place initiation at comparable depths, which is what matters practically, so the disagreement rarely changes a case-depth decision.

Surface traction moves the peak. Gear contact is rolling–sliding, so a tangential traction µp acts along the flank. As the traction coefficient rises, the shear maximum migrates toward the surface; above roughly µ ≈ 0.25–0.30 the maximum sits at the surface. This is the mechanical reason that high-sliding, poorly lubricated regions — tip and root of the sun pinion — fail from the surface inward, while well-lubricated pitch-line regions fail from below.

Finally, the field decays slowly. At z = 2b the shear is still ~70 % of its peak; at z = 4b, ~40 %. Hardness, by contrast, can fall from case to core over a much shorter distance. That mismatch is the whole case-depth problem.

4. Taxonomy of the damage modes

The four modes below are frequently reported as "pitting" indiscriminately. They have different initiation depths, different drivers, and different remedies. Terminology follows ISO 10825-1¹⁰ and ANSI/AGMA 1010-F14⁹.

4.1 Initial (run-in) pitting

Small, shallow pits appearing early in life, usually just below the pitch line, where local load concentration exceeds local strength before the flanks have conformed. It is characteristically self-arresting: the pits redistribute load, the surface work-hardens and runs in, and progression stops. Distinguishing arrested initial pitting from early progressive pitting is a matter of trend, not of a single inspection — this is a real limitation of one-time visual assessment.

4.2 Progressive macropitting

Craters typically 0.3–2 mm across that grow in number and area, produced by cracks initiating at or just below the surface and propagating back to it. Pits are often fan-shaped with the apex pointing against the direction of surface friction. Progression is not linear: liberated pit edges act as stress raisers and debris dents adjacent flanks, so the damage rate accelerates.

4.3 Micropitting (gray staining)

Asperity-scale contact fatigue. Cracks initiate at asperity contacts within a few micrometers of the surface and produce pits on the order of 10–20 µm, which collectively read to the eye as a dull gray or frosted band. It is a lubrication-regime failure, driven by low specific film thickness (§6) rather than by bulk contact pressure, and its main consequence is loss of profile accuracy — which then raises dynamic load and can seed macropitting. ISO/TS 6336-22⁶ provides a calculation method based on local film thickness.

4.4 Subcase fatigue and spalling

Initiation deep below the flank, in the case–core transition, where the applied shear stress has not decayed as fast as hardness has fallen (Figure 2). Cracks propagate roughly parallel to the surface before turning up, liberating large plates of case material. The distinguishing features are scale (spalls are typically several millimeters and can span a large fraction of the active flank) and abruptness — subcase spalling gives far less warning than progressive pitting. Where a single deep-initiated crack propagates across the tooth rather than back to the flank, the result is tooth flank fracture, for which ISO/TS 6336-4³ provides a calculation method. The remedies are metallurgical and geometric — deeper effective case, higher core hardness, cleaner steel, larger R — not lubrication.

5. Case depth and the hardness gradient

Carburized and hardened gearing is commonly specified with a surface hardness in the 58–62 HRC range and a core in the region of 30–45 HRC, with a compressive residual stress in the case typically quoted in the range of a few hundred MPa. The compressive residual stress is a genuine contributor to flank durability — it must be overcome before a crack can open — and it is lost if the part is reground into or beyond the effective case.

Effective case depth is widely specified as a proportion of normal module, commonly cited in the range 0.15–0.25 × m_n, with the higher end used for high contact stress or where subcase margin is a concern. ISO 6336-5⁴ and AGMA 923-B05¹³ give the material-quality framework in which such specifications sit. Two practical cautions apply:

• The measurement limit differs between conventions. ISO 18203¹⁴ defines effective case depth for a carburized case to a limiting hardness of 550 HV; long-standing North American practice per SAE J423¹⁵ commonly uses 50 HRC (≈ 513 HV). A part can be compliant to one and marginal to the other. Always state the limit hardness with the depth.
• A module-based rule is a proxy, not a criterion. The real criterion is that permissible shear exceeds applied shear at every depth (Figure 2). Because the two curves run nearly parallel through the transition zone, the design margin is thinnest there, not at the surface — so a modest overload, an unexpectedly low core hardness or a shallow case can flip the comparison over a broad depth band at once.

Hardness profiles are measured by microindentation on a sectioned tooth per ASTM E384¹⁸ or ISO 6507-1¹⁷, taken normal to the flank at a defined position on the profile; results taken at the tooth tip and at mid-flank are not interchangeable.

6. Lubrication: film thickness and the lambda ratio

Gear flanks operate in the elastohydrodynamic (EHL) regime. Established EHL solutions (Dowson & Higginson; Hamrock & Dowson) give minimum film thickness for line contact in the approximate form

where U is entrainment velocity and α the pressure–viscosity coefficient (typically ~10–25 GPa⁻¹ for common gear base oils). The exponents carry the message: film thickness is governed by speed and oil and is almost indifferent to load. A drive that is adequately lubricated at travel speed can be in near-boundary contact during slow, high-torque creep — which is precisely the duty a travel or swing drive sees most.

Durability is judged by the specific film thickness λ = h_min/σ_c. The conventional interpretation is λ > 2–3 full film, 1 < λ < 2 mixed, λ < 1 predominantly boundary with a high micropitting risk. Because σ_c sits in the denominator, surface finish is as powerful a lever as viscosity: moving from a ground finish (R_a of order 0.3–0.8 µm) to a superfinished flank can raise λ several-fold without changing the oil. ISO 6336-2² does not use λ directly, instead applying lubricant, velocity and roughness factors (Z_L, Z_v, Z_R) to the permissible stress; ISO/TS 6336-22⁶ works with local film thickness explicitly.

Contamination belongs in the same discussion. A hard particle larger than h_min — that is, larger than a fraction of a micrometer — will dent the flank, and the raised shoulder of the dent is a stress concentration that initiates pitting. Specifying and verifying fluid cleanliness to a code such as ISO 4406¹⁹ is therefore a durability measure, not a housekeeping one.

7. Surface durability rating: the shape of the method

ISO 6336-2² and ANSI/AGMA 2001-D04⁷ differ in notation and in some factor definitions, but share a structure worth understanding independently of either document:

1. Compute a nominal contact stress from transmitted load and mesh geometry, essentially the Hertz expression of §2.2 dressed with geometry factors (zone factor, contact ratio factor, helix angle factor).
2. Inflate it by a chain of load-modifying factors: application factor (K_A) for external duty variation, dynamic factor (K_v) for internal excitation, face and transverse load distribution factors (K_Hβ, K_Hα) for how unevenly the load actually sits on the flank, and — for epicyclic stages — the mesh load factor K_γ (§8).
3. Compare against a permissible stress derived from an endurance limit σ_Hlim for the material and quality grade (ISO 6336-5⁴ quotes values for case-hardened wrought steel commonly in the 1300–1650 MPa band across its ML/MQ/ME grades) modified by life, lubrication, roughness, velocity, work-hardening and size factors.
4. Divide by a safety factor chosen for the consequences of failure.

The life factor deserves attention because it explains most of the field behavior. In ISO 6336-2² the pitting life factor Z_NT for case-carburized steel falls from about 1.6 at 10⁵ cycles to 1.0 at 5×10⁷. Fitting a power law to those two points gives σ_H ∝ N^−0.076, i.e. N ∝ σ_H⁻¹³. Since σ_H ∝ √w′, life varies roughly as the inverse sixth to seventh power of load. An illustrative consequence: a sustained 10 % torque overload costs roughly half the pitting life; a 25 % overload costs about three quarters. (This exponent is a derivation from the shape of the published curve for illustration, not a measured result, and the real curve is not a pure power law.) It also explains why cycle counting matters: in a planetary set a sun tooth meshes once per planet per revolution relative to the carrier, while a planet flank meshes once per planet revolution, and relative to the carrier the planet turns at the speed of the sun times the ratio of sun to planet tooth count. With four planets a sun tooth therefore accumulates four times the cycles of any one planet flank, multiplied by the ratio of planet to sun tooth count: four times in a fixed-ring 4:1 stage, where the two counts are equal, and six times at 5:1.

8. Load sharing between planets

A planetary stage is only as good as its worst-loaded planet. Ideal analysis assumes the input torque divides equally among n planets; reality depends on whether the system can find that equilibrium.

With three planets and one floating member (a sun free to translate radially, or a floating ring or carrier), the assembly is statically determinate: three coplanar forces at 120° can equilibrate a floating sun in exactly one way, and that way is equal sharing. Small position errors are absorbed by sun translation. With four or more planets, or with all members rigidly located, the system is over-constrained and load sharing is decided by elastic compliance and by the tolerance stack — planet pin position and parallelism, carrier bore spacing, tooth thickness variation, runout and bearing clearance.

The scale problem is that the mesh is very stiff, and a rigidly located carrier has little compliance to set against it. ISO 6336-1¹ quotes representative values of single tooth-pair stiffness c′ ≈ 14 N/(mm·µm) and mean mesh stiffness c_γ ≈ 20 N/(mm·µm). On a 60 mm face width, closing a 20 µm positional error through the mesh alone would require on the order of 24 kN. Nothing in a rigidly located carrier is that soft, so the error is not absorbed — it appears as load imbalance. This is the engineering rationale for deliberately compliant elements: floating suns, flexible pins, and the flexible ring gears used in some designs, all of which introduce a compliance comparable to, or softer than, the mesh so that geometry errors can be accommodated.

Standards handle this with a mesh load factor K_γ applied directly to the load. ANSI/AGMA 6123⁸ tabulates values by planet count and by whether a floating member is present; published practice puts three-planet floating arrangements near 1.00–1.05 and rigidly located sets of four or more planets appreciably higher. Combined with the load–life exponent of §7, the consequence is severe: a K_γ of 1.25 means the worst planet mesh reaches its pitting life roughly four times sooner than an equal-sharing calculation predicts. Gear accuracy grade (ISO 1328-1¹¹, or ANSI/AGMA 2015-1-A01¹²) and carrier machining tolerance are therefore durability parameters, not merely quality parameters.

The same argument applies across the face width. K_Hβ captures misalignment from carrier deflection, pin bending under load and housing distortion; lead crowning and helix modification exist to keep the contact from migrating to one end of the flank, where the effective w′ — and hence σ_H — can be far above nominal.

9. Limits of the method

The rating standards are calibrated correlations, not first-principles predictions. They were developed largely from test gearing under controlled conditions and carry substantial scatter — flank durability data typically show a wide band between 10 % and 90 % failure probability, and the standards are explicit that their permissible stresses correspond to a stated failure probability, not to a guarantee. They also assume a load spectrum that must be supplied honestly; ISO 6336-6⁵ covers accumulation under variable load, but no method can compensate for an understated duty cycle. Where a design is governed by subsurface initiation, steel cleanliness — the inclusion population that seeds deep cracks — becomes a first-order variable that the conventional surface durability calculation does not see at all, which is why ISO/TS 6336-4³ exists as a separate assessment.