| 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
- Assume initial flaw size a_i from NDE capability (a_i = resolution limit)
- Calculate a_cr from K_IC
- Calculate N_f (cycles from a_i to a_cr)
- Set inspection interval = N_f/2 (safety factor 2)
- 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.
