contact-mechanics

name	contact-mechanics
description	Contact mechanics — Hertz contact stress/deformation for sphere, cylinder, elliptical; JKR/DMT adhesion models; rough surface contact (GW model); fretting, stick-slip zones; elastic-plastic contact.
metadata	{"priority":7,"promptSignals":{"phrases":["contact mechanics","Hertz contact","contact stress","contact deformation","JKR model","fretting contact"],"minScore":3}}

Contact Mechanics — Complete Skill

Hertz Contact Theory (Elastic)

Assumptions

• Elastic, homogeneous, isotropic materials
• Contact area << body dimensions (half-space assumption)
• Frictionless contact
• Small strains; no adhesion

Sphere-Sphere (or Ball on Race) — Point Contact

Effective radius: 1/R = 1/R₁ + 1/R₂ [same curvature sign; for convex bodies] For sphere on flat: R = R_sphere For sphere in spherical socket: 1/R = 1/R₁ - 1/R₂ (opposite sign → conforming; R larger)

Reduced modulus: 1/E* = (1-ν₁²)/E₁ + (1-ν₂²)/E₂

Contact radius: a = (3PR / 4E*)^(1/3) [m]

Normal approach (mutual approach): δ = a² / R = (9P² / (16RE*²))^(1/3) [m]

Peak contact pressure: p₀ = 3P / (2πa²) [Pa; average = 2/3 p₀]

Pressure distribution: p(r) = p₀ × (1 - r²/a²)^(1/2) [Hertzian ellipsoid]

Subsurface stresses on axis of symmetry (z-axis): σ_z = -p₀ × (1 + z/a × arctan(a/z))⁻¹ [compressive; approximate] τ_max ≈ 0.31 p₀ at z ≈ 0.48a [max shear stress depth]

Elastic stiffness (contact stiffness): K = dP/dδ = 2Ea = 2E(3PR/(4E*))^(1/3) [N/m; increases with load]

Cylinder-Cylinder — Line Contact

Effective radius: 1/R = 1/R₁ + 1/R₂

Contact half-width: b = (4PR / (πLE*))^(1/2) [m; L = contact length; P = total load]

Peak pressure: p₀ = 2P / (πbL) [Pa]

Pressure distribution: p(x) = p₀ × (1 - x²/b²)^(1/2)

Subsurface stresses (line contact): τ_max = 0.30 p₀ at z = 0.786b τ_orth = 0.25 p₀ at z = 0.5b [orthogonal shear; key for fatigue]

Elliptical Contact (General Case)

Applicable when contact ellipse is not circular (misaligned cylinders, toroidal bodies): k = a_ellipse / b_ellipse [ellipticity ratio; depends on geometry] Requires complete elliptic integrals K(e), E(e) for exact solution

Approximate (Brewe-Hamrock, 1977): k ≈ 1.0339 × (R_y/R_x)^0.636 F = 1.5277 + 0.6023 × ln(R_y/R_x) E_e = 1.0003 + 0.5968 × (R_x/R_y) a = (6kE_e PR_x / (πFE*))^(1/3) b = a/k; p₀ = 3P/(2πab)

Adhesion Models

JKR Model (Johnson-Kendall-Roberts) — soft, high adhesion

Pull-off force: P_pull-off = -(3/2)π × w × R_eff [N; w = work of adhesion [J/m²]] Contact radius under load P: a³ = (R/E*) × [P + 3πwR + √(6πwRP + (3πwR)²)]

Work of adhesion: w = γ₁ + γ₂ - γ₁₂ [J/m²; surface energies] Typical: rubber-glass w ≈ 0.06 J/m²; steel-steel (in air) w ≈ 1–2 J/m²

DMT Model (Derjaguin-Muller-Toporov) — hard, low adhesion

Pull-off force: P_pull-off = -2π × w × R_eff [N] Contact area follows Hertz; van der Waals forces outside contact

JKR vs. DMT transition (Maugis parameter): λ = 2σ₀(R_eff/πwE*²)^(1/3) [σ₀ = adhesion stress] λ >> 1: JKR valid (soft, large tip); λ << 1: DMT valid (stiff, small tip) λ ≈ 1: Maugis-Dugdale (numerical)

Rough Surface Contact (Greenwood-Williamson Model)

Real contact area A_real << A_apparent: A_real / A_apparent ≈ E* / (H × σ_s / R_s^(1/2)) [H = hardness; σ_s = RMS roughness; R_s = asperity radius]

GW model: Total force P = N_asperity × E* × R_s^(1/2) × ∫ (z-d)^(3/2) φ(z) dz d = separation; φ(z) = Gaussian height distribution

Plasticity index: ψ = (E*/H) × (σ_s/R_s)^(1/2) ψ < 0.6: elastic asperity contact; ψ > 1.0: plastic asperity contact (real surfaces typically 1–10)

Sliding Contact (Friction and Stick-Slip)

Tangential loading with friction (Mindlin, 1953): Applied tangential force T < μP → partial slip at edge; stick zone in center Full-slip (Coulomb limit): T = μP

Stick radius: c/a = (1 - T/μP)^(1/3) [c = stick zone radius; a = Hertz contact radius]

Fretting wear (Archard for fretting): V = k_f × P × s × n [k_f = fretting wear coefficient; s = slip amplitude; n = cycles] Fretting damage occurs for s > 2–5 μm; ceases for s < 1 μm (full stick)

Fretting fatigue life reduction: σ_f_fretting ≈ 0.5–0.7 × σ_f_plain (fatigue limit reduced 30–50% by fretting)

Elastic-Plastic Contact

Initial yield criterion: p₀ / H_V ≈ 1.07 → first yield at p₀ ≈ 1.07H_V (Vickers hardness → yield) For steel: H_V [MPa] ≈ 3σ_y → p₀_yield = 1.07 × 3σ_y ≈ 3.2σ_y

Fully plastic contact: p₀_plastic ≈ 3σ_y = H (hardness) Load: P = H × πa² [H in Pa; a = contact radius — independent of E*]

Intermediate regime (Biwa-Storakers): c/a (elastic-plastic boundary) from numerical or empirical models

Contact hardness (Meyer): H_m = 4P / (πd²) [d = indenter diameter; P = load — hardness testing basis]

Thick Coatings / Layered Systems

For hard coating (DLC, TiN) on softer substrate: Effective E* depends on h_coat/a ratio If a >> h_coat: substrate dominates → E_eff → E_substrate If a << h_coat: coating dominates → E_eff → E_coating

Critical load for coating failure (Buckle criterion): P_crit ≈ 3H_c × t² [H_c = coating hardness; t = coating thickness; contact radius ≤ 0.1t]

Standards

Standard	Scope
ASTM E384	Microindentation hardness (Vickers, Knoop)
ISO 14577	Instrumented indentation — Oliver-Pharr method
ASTM G143	Fretting wear test
ISO 4516	Vickers and Knoop hardness of metallic coatings

Output

Provide: contact type (point/line/elliptical), effective radius R [mm], reduced modulus E* [GPa], contact radius a (or half-width b) [mm], peak Hertz pressure p₀ [MPa] vs. material hardness limit, normal approach δ [μm], depth of maximum shear stress z_τmax [mm], contact stiffness K [N/μm], adhesion model (JKR/DMT/none) with pull-off force [N], Λ ratio (if sliding/fatigue relevant), fretting slip amplitude s [μm] and damage regime, elastic-plastic transition check (p₀ vs. 3σ_y), and applicable standard (ISO 14577, ASTM G143).