| name | contact-stress |
| description | Hertzian contact stress — sphere-sphere, cylinder-cylinder, sphere-flat, cylinder-flat, general elliptical. Contact pressure, contact area, subsurface shear, pitting failure, bearing/gear applications. |
| metadata | {"priority":7,"promptSignals":{"phrases":["contact stress","Hertz","hertzian","contact pressure","ball contact","roller contact","pitting","contact area"],"minScore":4}} |
Hertzian Contact Stress — Complete Skill
Fundamental Hertz Theory
Contact between curved bodies produces:
- Finite contact area (circle or ellipse, not a point/line)
- Maximum pressure at center
- Subsurface maximum shear stress (causing pitting fatigue)
Key assumptions:
- Elastic bodies (no yielding)
- Smooth surfaces (no friction — frictionless contact)
- Small contact area compared to radii of curvature
- Homogeneous, isotropic materials
Combined Elastic Modulus
E* = ((1-ν₁²)/E₁ + (1-ν₂²)/E₂)^(-1)
For identical materials (E₁=E₂=E, ν₁=ν₂=ν): E* = E/(2(1-ν²))
For steel (E=200GPa, ν=0.3): E* = 109.9 GPa
1. Sphere on Sphere (3D — Circular Contact)
Contact radius:
a = (3F(1/R₁+1/R₂)^(-1) / (4E*))^(1/3)
Let R* = (1/R₁ + 1/R₂)^(-1) [combined radius]:
a = (3F·R* / (4E*))^(1/3)
Special: sphere on flat (R₂=∞): R* = R₁
Identical spheres: R* = R/2
Maximum contact pressure:
p_max = 3F/(2πa²) = (6FE²/(π³R²))^(1/3)
Contact area: A = πa²
Stress distribution (along z-axis):
σ_x = σ_y = -p_max·[(1-z/a·arctan(a/z))·(1+ν) - 1/(2(1+(z/a)²))]
σ_z = -p_max/(1+(z/a)²)
τ_max = 0.31·p_max at depth z ≈ 0.48a (subsurface max shear — pitting origin)
2. Cylinder on Cylinder (2D — Line Contact)
Half-contact width:
b = (4F·R*/(π·E*·L))^(1/2)
Where: 1/R* = 1/R₁ + 1/R₂ (parallel cylinders same sign, crossed = subtract)
L = contact length
Special: cylinder on flat (R₂=∞): R* = R₁
Maximum contact pressure:
p_max = 2F/(πbL) = (F·E*/(π·R*·L))^(1/2)
Stress along z-axis (below centerline):
τ_max = 0.30·p_max at depth z ≈ 0.786b (subsurface max shear — pitting origin)
σ_x at surface = -2ν·p_max (principal stress parallel to surface)
σ_z at surface = -p_max
3. General Elliptical Contact
For bodies with principal curvatures in two planes:
Σρ = 1/R₁_x + 1/R₁_y + 1/R₂_x + 1/R₂_y
Contact ellipse a (major) × b (minor)
Requires elliptic integrals (k, e) from Hertz tables vs. cos(θ) parameter
For gears: Use AGMA contact stress formula (which accounts for geometry factor I/Z_H)
4. Ball in Groove (Bearing)
Inner race: 1/R* = 1/d_ball - 1/(f_i·d_ball) (f_i = 0.51-0.53, groove radius factor)
Outer race: 1/R* = 1/d_ball + 1/(f_o·d_ball)
Contact stress in deep-groove ball bearings typically 1500-3000 MPa at rated load — high but in compressive hydrostatic state.
5. Rail/Wheel (Specialized)
Wheel: R₁ = 0.5m (typical), Rail: R₂ ≈ 0.3m (crown)
F = 100-200 kN (heavy freight)
p_max typically 800-1500 MPa → surface/subsurface fatigue over time
Failure: Pitting Criterion
Static: p_max < σ_y (Hertz contact is triaxial compression, effectively multiply by ~0.6)
Effectively: no plastic yielding if p_max < 3·k where k = Sy/2 (Tresca)
Onset of yield: p_max = 1.60·Sy (ball) or 1.67·Sy (cylinder) — Brinell test basis
Fatigue pitting: cyclic loading → subsurface shear fatigue at τ_max depth
Allowable contact stress from AGMA/bearing life tables
For gears: σ_c,allow = S_c·Z_N/(K_T·K_R) — compare to AGMA contact stress
Design Recommendations
- Use larger radii (reduce p_max)
- Increase hardness (raise yield and fatigue limit)
- Use full-film EHL lubrication (reduces asperity stress)
- Avoid edge loading (chamfer or crown edges)
- Shot peen (compressive residual stress to depth of max shear)
- Surface coating (nitriding raises surface hardness to 65+ HRC)
Output
Provide: contact geometry type, a or b [mm], p_max [MPa], location of τ_max, pitting life assessment, design recommendation.
