contact-stress

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.