| name | paris-law |
| description | Paris law fatigue crack growth — da/dN = C(ΔK)^m, stress intensity factor range ΔK, threshold ΔK_th, crack growth regimes (I/II/III), retardation (Willenborg, Wheeler), short crack regime, mixed-mode K_I/K_II/K_III, fracture toughness K_Ic at instability, ASTM E647 (FCG testing), inspection interval calculation, damage-tolerant design (LEFM), and aerospace/pressure vessel applications. |
| metadata | {"priority":7,"promptSignals":{"phrases":["Paris law","fatigue crack growth","crack growth rate","da/dN","stress intensity range","damage tolerant design"],"minScore":3}} |
Paris Law Fatigue Crack Growth — Complete Skill
Fracture Mechanics Fundamentals
Stress Intensity Factor (SIF)
Mode I (opening):
K_I = Y × σ × √(π × a) [MPa√m; Y = geometry factor; σ = nominal stress [MPa]; a = crack half-length [m]]
Geometry factors Y (selected cases):
Through crack in infinite plate: Y = 1.0
Edge crack in semi-infinite plate: Y = 1.12
Through crack in finite width plate (W): Y = √(sec(πa/W)) × correction [Feddersen; a/W < 0.7]
Surface semi-elliptical crack: Y = (1/Q) × F_s × F_w [Q = shape factor from ellipse ratio a/c; F_s, F_w = surface and finite-width correction]
Stress intensity for common geometries (ASTM E399/E647):
Compact tension (CT) specimen: K = P × f(a/W) / (B × W^0.5)
f(a/W) = (2+a/W)(0.886 + 4.64(a/W) - 13.32(a/W)² + 14.72(a/W)³ - 5.6(a/W)⁴) / (1 - a/W)^1.5
Superposition:
Combined loading: K_total = K₁ + K₂ [linear superposition of K_I from each load component]
Mixed mode: KI from membrane + KI from bending, etc.
Fracture Toughness K_Ic
Plane-strain fracture toughness K_Ic:
Critical K_I for unstable fracture onset under plane strain (most conservative)
Units: MPa√m = MN·m^(-3/2)
K_Ic for common materials:
| Material | K_Ic (MPa√m) |
|---|
| 4340 steel (high strength) | 50–65 |
| 300M steel (aerospace) | 60–90 |
| 7075-T6 aluminum | 28–32 |
| 2024-T3 aluminum | 34–44 |
| Ti-6Al-4V (STA) | 55–110 |
| Inconel 718 | 120–145 |
| PMMA (acrylic) | 0.7–1.0 |
| SiC ceramic | 3–4 |
Validity (ASTM E399):
Plane strain condition: B ≥ 2.5 × (K_Ic/σ_y)²; a ≥ 2.5 × (K_Ic/σ_y)²
If invalid: use J_Ic or K_eff (plane stress; higher apparent toughness)
Failure criterion:
K_I ≥ K_Ic → unstable fracture (critical crack size)
a_crit = (1/π) × (K_Ic / (Y × σ))² [critical crack size for given stress]
Paris Law
Regime II Crack Growth
Paris-Erdogan law (1963):
da/dN = C × (ΔK)^m [da/dN = crack growth per cycle [m/cycle]; ΔK = K_max - K_min [MPa√m]; C, m = material constants]
ΔK definition:
ΔK = Y × Δσ × √(πa) [Δσ = σ_max - σ_min per cycle]
R ratio: R = K_min/K_max = σ_min/σ_max [R = 0 for tension-only; R < 0 for reversed]
ΔK_eff = ΔK × (1-R)^(1-m_exp) corrections apply for R > 0 (Walker, Forman)
Units note:
C in (MPa√m)^(-m) × m/cycle
da/dN in m/cycle (or mm/cycle in some tables); be consistent
Paris constants (common materials in air):
| Material | C [MPa√m, m/cycle] | m |
|---|
| A533B steel | 6.9×10⁻¹² | 3.0 |
| 4340 steel (high strength) | 1.0×10⁻¹¹ | 3.26 |
| 7075-T6 Al (L-T) | 3.5×10⁻¹¹ | 3.43 |
| 2024-T3 Al | 5.6×10⁻¹² | 3.02 |
| Ti-6Al-4V (R=0.1) | 2.0×10⁻¹² | 3.5 |
| Austenitic stainless (SS316) | 4.0×10⁻¹² | 3.0 |
Three Regions of Crack Growth
Region I (near-threshold):
ΔK < ΔK_th → no measurable growth (< 10⁻¹⁰ m/cycle)
ΔK_th (threshold): 4–12 MPa√m (steel); 2–4 MPa√m (aluminum)
ΔK_th decreases with R: ΔK_th = ΔK_th,0 × (1-R)^γ [γ ≈ 0.5–1.0]
Region II (Paris/power-law):
10⁻⁸ < da/dN < 10⁻⁶ m/cycle; da/dN = C(ΔK)^m
Linear on log-log plot; Paris law valid here
Engineering "safe" range: most fatigue crack growth occurs in Region II
Region III (near-fracture):
ΔK approaches K_Ic/√(1-R); rapid acceleration → instability
Modified Paris: da/dN = C × (ΔK)^m / (1 - ΔK/ΔK_c) [Forman equation]
K_max → K_Ic → unstable fracture (final cycle)
Walker and Forman Equations
Walker equation (R-ratio correction):
da/dN = C_W × [ΔK / (1-R)^(1-m_W)]^m_W [accounts for mean stress (R ratio)]
C_W, m_W = Walker constants (fit from data at multiple R ratios)
Reduces to Paris at R = 0: da/dN = C_W × ΔK^m_W
Forman equation (Regions II + III):
da/dN = C_F × ΔK^m_F / [(1-R) × K_Ic - ΔK]
Predicts growth rate acceleration as ΔK → (1-R) × K_Ic
C_F, m_F from curve fit; asymptote at ΔK → (1-R) K_Ic → da/dN → ∞
Crack Growth Integration
Numerical Integration (Cycle-by-Cycle)
Procedure:
- Initial crack size a₀ (from NDE detectability or assumed flaw)
- Applied stress Δσ(N) (constant amplitude or load spectrum)
- For each cycle increment ΔN:
a. Compute ΔK_n = Y × Δσ × √(π × a_n)
b. da/dN_n = C × (ΔK_n)^m
c. Δa = (da/dN_n) × ΔN
d. a_(n+1) = a_n + Δa
- Continue until a_n ≥ a_crit (K_max → K_Ic) → failure
- N_failure = total cycles
Python integration:
import numpy as np
C = 6.9e-12
m = 3.0
K_Ic = 200
Y = 1.12
delta_sigma = 100
a0 = 0.001
a_crit = (1/np.pi) * (K_Ic / (Y * delta_sigma))**2
a = a0
N = 0
while a < a_crit:
dK = Y * delta_sigma * np.sqrt(np.pi * a)
da_dN = C * dK**m
da = da_dN * 1
a += da
N += 1
print(f"N_failure = {N:,} cycles; a_final = {a*1000:.2f} mm")
Analytical integration (constant amplitude, constant Y):
da/dN = C × (Y × Δσ × √π)^m × a^(m/2)
Integrate: ∫ a^(-m/2) da = C × (Y × Δσ × √π)^m × N
For m ≠ 2: N_failure = [a_f^(1-m/2) - a₀^(1-m/2)] / [(1-m/2) × C × (Y Δσ √π)^m]
For m = 2: N_failure = ln(a_f/a₀) / [C × (Y Δσ √π)²]
Approximate (for m = 3, simplification):
N ≈ 2 / [(m-2) × C × (Y Δσ √π)^m × a₀^(m/2-1)] [dominant term when a₀ << a_f]
Retardation Effects
Overload Retardation
Physical mechanism:
Single tensile overload creates large plastic zone → compressive residual stress → subsequent cycles partially closed → reduced ΔK_eff → crack growth retards
Willenborg model:
Effective K_max reduced by K_res (residual stress effect)
K_max_eff = K_max - K_res [K_res proportional to overload ratio and current crack position in plastic zone]
ΔK_eff = K_max_eff - max(K_min, 0) [tensile only; compressive K does not drive crack]
Wheeler model:
Retardation factor φ: da/dN_retarded = φ × da/dN_Paris
φ = (a_pl_n / r_pl_OL)^p [a_pl_n = current crack tip to OL plastic zone boundary; r_pl_OL = OL plastic zone radius; p = shaping exponent fitted from data]
φ = 1 (no retardation) to φ < 1 (retarded); φ → 1 as crack grows through plastic zone
NASGRO equation (NASA):
da/dN = C_N × [(1-f)/(1-R)]^n × ΔK^p / [(1-ΔK_th/ΔK)^q × (1-K_max/K_Ic)^(1-n)]
Includes: threshold; fracture instability; crack closure (f = crack closure function)
Database: NASGRO 9 material library (aerospace alloys); used in NASGRO software (SWT)
Inspection Interval Calculation
Damage-Tolerant Design
Inspection interval (half-life approach):
N_fail from a₀ (minimum detectable flaw) to a_crit (critical crack size)
Inspection interval = N_fail / 2 (detect crack before it reaches critical size)
More precisely: N_interval = N_fail_detectable [from a_inspect to a_crit with specified confidence]
Probability of detection (POD):
NDE method gives POD vs. crack size curve
a_90/95: crack size detectable at 90% POD with 95% confidence
Use a₀ = a_90/95 as initial crack size for inspection interval calculation
Two-inspection philosophy:
If inspection interval = I: first inspection at N = I; second at N = 2I; etc.
No crack found at inspection → confidence that a < a_detect → reset clock
Crack found → repair; restart life
Example:
a₀ (minimum detectable, UT) = 5 mm; a_crit = 25 mm (plane strain fracture)
N_failure = 400,000 cycles (calculated); Inspection interval = 200,000 cycles
Schedule inspection every 200,000 cycles or every 2 years (whichever shorter)
ASTM E647 (FCG Testing)
Specimen: compact tension (CT); pre-cracked from fatigue; a₀/W = 0.25–0.60
Test method: constant load amplitude; measure crack length a vs. cycles N optically or with clip gauge
da/dN calculation: secant method da/dN ≈ (a_2 - a_1)/(N_2 - N_1) at ΔK = avg
K-decreasing test (ΔK_th): reduce ΔK as crack grows (constant normalized K gradient K⁻¹ dK/da = -0.08 mm⁻¹); measure ΔK_th when da/dN < 10⁻¹⁰ m/cycle
Data validity: K_max ≤ 0.8 K_Ic; a/W ≤ 0.60; no extensive plasticity (small-scale yielding)
Applications
Aerospace (FAA AC 25.571)
Damage tolerance requirement:
Aircraft structure must sustain loads with damage (cracking) for specified inspection interval
a₀: detectable crack from manufacturing (typically 1.27 mm = 0.05 in for bolted holes)
N_inspection: structural inspection interval (defined by maintenance program)
Requirement: a grows from a₀ to < a_crit over 2 × N_inspection (slow growth)
NASGRO/AFGROW: USAF Damage Tolerant Analysis tool; includes retardation; variable amplitude
F/A-18, B-777: fleet life tracked by Structural Analysis Monitoring; actual flight spectrum input
Pressure Vessels (ASME XI / API 579)
Nuclear RPV beltline crack growth:
da/dN = C_env × (ΔK)^m_env [environmental enhancement; LWR water]
C_env > C_air by factor 2–10× depending on temperature and dissolved oxygen
API 579-1 Fitness-For-Service:
Appendix D: fatigue crack growth assessment for in-service cracks
Level 1: conservative; Level 2: Paris law with API C, m values; Level 3: site-specific da/dN data
Standards and References
| Standard | Scope |
|---|
| ASTM E647 | Fatigue crack growth rate testing (da/dN vs. ΔK) |
| ASTM E399 | Plane-strain fracture toughness K_Ic |
| ASTM E1820 | J-integral fracture toughness |
| API 579-1 / ASME FFS-1 | Fitness-for-service flaw assessment |
| FAA AC 25.571 | Damage tolerance for aircraft structures |
| MIL-A-83444 | Aircraft damage tolerance requirements |
| NASGRO Database | Material FCG constants (aerospace alloys) |
Output
Provide: material (Paris constants C, m at R=0 [units]; K_Ic [MPa√m]; ΔK_th [MPa√m]), crack geometry (Y formula; initial crack a₀ [mm] from NDE; critical crack a_crit [mm] from K_Ic check), applied stress (Δσ [MPa]; R ratio; spectrum if variable), ΔK at a₀ and a_crit [MPa√m] (verify a₀ in Region II: ΔK > ΔK_th), cycle-by-cycle integration (or analytical formula result): N_failure [cycles], retardation considered (Willenborg/Wheeler? overload magnitude?), inspection interval [cycles or hours], minimum detectable flaw size a_90/95 from NDE method [mm], critical crack size from fracture: a_crit [mm] (back-calculate from K_Ic and operating stress), POD from NDE method [%], and applicable standard (ASTM E647, API 579, FAA AC 25.571).
