paris-law

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:

1. Initial crack size a₀ (from NDE detectability or assumed flaw)
2. Applied stress Δσ(N) (constant amplitude or load spectrum)
3. 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
4. Continue until a_n ≥ a_crit (K_max → K_Ic) → failure
5. N_failure = total cycles

Python integration:

import numpy as np

# Material and geometry
C = 6.9e-12  # (MPa*m^0.5)^-m * m/cycle (A533B steel)
m = 3.0
K_Ic = 200  # MPa*m^0.5
Y = 1.12    # edge crack
delta_sigma = 100  # MPa stress range
a0 = 0.001  # 1 mm initial crack
a_crit = (1/np.pi) * (K_Ic / (Y * delta_sigma))**2  # critical crack size

# Integration
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  # one cycle at a time (or use larger ΔN for efficiency)
    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).