fracture-mechanics

name	fracture-mechanics
description	Fracture mechanics — stress intensity factor KI, critical crack size, fracture toughness KIC, Paris law fatigue crack growth, LEFM vs. EPFM, J-integral, damage tolerance design.
metadata	{"priority":7,"promptSignals":{"phrases":["fracture","crack","stress intensity","KIC","fracture toughness","fatigue crack growth","Paris law","damage tolerance","crack propagation"],"minScore":4}}

Fracture Mechanics — Complete Skill

Linear Elastic Fracture Mechanics (LEFM)

Stress Intensity Factor: K_I = σ√(πa) · F(a/W) [MPa√m or ksi√in]

Where:

• σ = remote applied stress
• a = crack half-length (through crack) or depth (surface crack)
• F(a/W) = geometry correction factor (boundary effects)

Common geometry factors:

Geometry	F(a/W)
Through crack in infinite plate	1.0
Edge crack in semi-infinite plate	1.12
Through crack in finite width plate	sec(πa/2W)^0.5
Surface crack (semi-elliptical)	1.12·√(1/Φ) where Φ = elliptic integral
Center through crack in finite plate	F = (sec(πa/W))^0.5 (approximate)
Single edge notch bend (SENB)	F = complex function from ASTM

Fracture condition: K_I ≥ K_IC → unstable fracture

Fracture Toughness K_IC [MPa√m]

Material	K_IC
4340 steel (Q&T 260°C, Sy=1640MPa)	50
4340 steel (Q&T 425°C, Sy=1480MPa)	87
300M steel	68
7075-T651 Al	24
2024-T351 Al	36
Ti-6Al-4V	55
17-4 PH (H900)	60
Inconel 718	105
Alumina (Al₂O₃)	4
Silicon nitride (Si₃N₄)	6
Glass	0.7
Polymethylmethacrylate (PMMA)	1

LEFM validity (plane strain): a, B, (W-a) ≥ 2.5(K_IC/σ_y)² [specimen size requirement] If not met → plane stress (less conservative, use K_C instead of K_IC)

Critical Crack Size

a_cr = (1/π)(K_IC/(σ·F))² For edge crack: a_cr = (1/π)(K_IC/(1.12·σ))²

Design implication: Inspect for cracks; if a < a_cr/safety, part is safe.

Three Modes of Fracture

Mode I: Opening (tension normal to crack plane) — most dangerous Mode II: Sliding (in-plane shear) Mode III: Tearing (out-of-plane shear) Mixed mode: K_eff = √(K_I² + K_II² + K_III²/(1-ν))

Fatigue Crack Growth (Paris Law)

da/dN = C·(ΔK)^m

Where:

• da/dN = crack growth per cycle [m/cycle]
• ΔK = K_max - K_min = Δσ√(πa)·F [MPa√m]
• C, m = material constants

Typical values (steel): m ≈ 3-4 (units: m/cycle, MPa√m) C ≈ 6×10⁻¹² (m=3, SI units for steel)

Regions: Region I: threshold ΔK_th < ΔK → no growth (ΔK_th ≈ 5-10 MPa√m for steel) Region II: Paris law valid (linear on log-log) Region III: K_max → K_IC → rapid growth

Life prediction: N_f = ∫[a_i to a_f] da/(C·(Δσ√(πa)·F)^m)

For F=1, m≠2: N_f = (a_f^(1-m/2) - a_i^(1-m/2)) / (C·(Δσ√π)^m·(1-m/2))

For m=3: N_f = 2/(C·(Δσ)³·π^1.5) · (a_i^(-0.5) - a_f^(-0.5))

Elastic-Plastic Fracture Mechanics (EPFM)

Used when plastic zone is significant: r_p = (1/2π)(K_I/σ_y)² >> small

J-integral (Hutchinson-Rice-Rosengren): J_IC = K_IC²(1-ν²)/E [plane strain] J_IC = K_IC²/E [plane stress] Fracture when J ≥ J_IC

Crack Tip Opening Displacement (CTOD): δ = K_I²/(σ_y·E·m) (m = 1 plane stress, 2 plane strain) Fracture when δ ≥ δ_c (critical CTOD, from testing)

Stress Corrosion Cracking (SCC)

K_ISCC < K_IC (threshold in corrosive environment, much lower) Crack grows at K_I > K_ISCC (even statically) Most susceptible: high-strength steel in saltwater, sensitized SS in chloride Test per ASTM E1681

Damage Tolerance Design

1. Assume initial flaw size a_i from NDE capability (a_i = resolution limit)
2. Calculate a_cr from K_IC
3. Calculate N_f (cycles from a_i to a_cr)
4. Set inspection interval = N_f/2 (safety factor 2)
5. Repeat inspection, retire part when a > a_cr/2

NDE detection limits: Visual: ~1mm PT/MT: ~0.25mm surface UT: ~1mm (depends on geometry) Eddy current: ~0.5mm surface X-ray: ~1-2mm

Output

Provide: K_I [MPa√m], K_IC [MPa√m], safety factor n=K_IC/K_I, critical crack size a_cr, fatigue life N_f if cyclic, inspection interval recommendation.