stress-intensity-factor

name	stress-intensity-factor
description	Stress intensity factor (SIF) — Mode I/II/III K_I/K_II/K_III, Irwin K-G relationship, SIF for common geometries (edge crack, central crack, surface crack, corner crack), stress intensity correction factors (F-functions, finite width correction), superposition principle, mixed-mode SIF (Erdogan-Sih criterion), K_Ic fracture toughness, CTOD, J-integral, linear elastic fracture mechanics (LEFM) validity, weight function method, and ASTM E399/E1820.
metadata	{"priority":7,"promptSignals":{"phrases":["stress intensity factor","fracture mechanics","K_I stress intensity","LEFM","fracture toughness","K_Ic"],"minScore":3}}

Stress Intensity Factor (SIF) — Complete Skill

Linear Elastic Fracture Mechanics (LEFM) Foundation

Stress Field Near Crack Tip

Mode I (opening, tension perpendicular to crack): σ_ij = (K_I / √(2πr)) × f_ij(θ) + O(r^0) [K_I = stress intensity; r = distance from crack tip; θ = angle from crack plane]

Mode II (in-plane shear): σ_ij = (K_II / √(2πr)) × g_ij(θ)

Mode III (anti-plane shear/tearing): τ_ij = (K_III / √(2πr)) × h_ij(θ)

Superposition: for mixed loading, K_I, K_II, K_III each calculated independently and summed for each mode

Physical significance: K_I quantifies the stress field intensity at the crack tip Units: MPa√m or ksi√in (1 ksi√in = 1.099 MPa√m) Fracture occurs when K_I ≥ K_Ic (mode I fracture toughness)

SIF for Common Geometries

Central Through-Crack in Infinite Plate (Griffith)

K_I = σ × √(π × a) [σ = remote applied stress; a = half-crack length] Basis: Inglis stress concentration + Griffith energy balance

Finite Width Correction

K_I = σ × √(π × a) × F(a/W) F(a/W) = Feddersen correction: F(a/W) = √(sec(π × a/W)) = √(sec(π × a/2W)) [depending on convention; a = half crack; W = half width] Or: F(a/W) = (1 - 0.025(a/W)² + 0.06(a/W)⁴) × √(sec(π × a/(2W))) [more accurate; Feddersen 1966]

Example: σ = 100 MPa; a = 20 mm = 0.02 m; W = 100 mm (half-width 50 mm → a/W = 0.40) K_I = 100 × √(π × 0.02) × √(sec(π × 0.02/(2×0.05))) = 100 × 0.2507 × √(sec(0.628)) = 100 × 0.2507 × √(1.236) = 100 × 0.2507 × 1.112 = 27.9 MPa√m

Edge Crack in Semi-Infinite Plate

K_I = 1.12 × σ × √(π × a) [a = crack depth from free surface; 1.12 = free surface correction factor] More precisely: K_I = F(a/W) × σ × √(π × a) F(a/W) = 1.12 - 0.231(a/W) + 10.55(a/W)² - 21.72(a/W)³ + 30.39(a/W)⁴ [Tadashi and Mura series]

Surface (Semi-Elliptical) Crack — Raju-Newman

Most common practical crack — on a surface, partially through: a = crack depth; 2c = crack surface length; t = plate thickness

SIF at deepest point: K_I = (σ × √(π × a) / Q^0.5) × F_surface [Q = shape factor; F_surface = boundary correction]

Q shape factor: Q = Φ² - 0.212 × (σ/σ_y)² [Φ = complete elliptic integral; usually Φ² ≈ 1 + 4.593 × (a/c)^1.65] For a/c = 0.5 (half-circle): Φ² = 2.464; Q = 2.464 - 0.212 × (σ/σ_y)² ≈ 2.464 (for σ/σ_y = 0)

Raju-Newman correction (1984): F(a/c, a/t, θ, φ) — function of crack shape ratio a/c, depth ratio a/t, location angle φ, and stress Tables and polynomial expressions in NASA CR-165807 and NASGRO documentation

At surface endpoint (φ = 0): K_I(surface) = 1.12 × σ × √(π × a / Q) [surface factor 1.12; slightly different from deepest point]

Through-Thickness Crack — Compact Tension (CT)

CT specimen (ASTM E399): K_I = P/(B×√W) × f(a/W) [P = applied force; B = specimen thickness; W = specimen width] f(a/W) = (2 + a/W) × (0.886 + 4.64(a/W) - 13.32(a/W)² + 14.72(a/W)³ - 5.60(a/W)⁴) / (1-a/W)^(3/2) Valid: 0.2 ≤ a/W ≤ 1.0

SENB (Single Edge Notched Bend): K_I = P × S / (B × W^(3/2)) × f(a/W) [S = span; f(a/W) = ASTM polynomial] f(a/W) = 3(a/W)^0.5 × [1.99 - (a/W)(1-a/W)(2.15 - 3.93(a/W) + 2.7(a/W)²)] / (2(1+2a/W)(1-a/W)^(3/2))

Pressurized Crack / Thermal Gradient

Internal pressure p in crack faces: K_I = p × √(π × a) [same form as remote tension; crack opens under pressure] Additive with remote K_I from far-field stress

Thermal gradient K_I (weight function): K_I = ∫₀ᵃ σ_T(x) × m(x,a) dx [m(x,a) = weight function for geometry; σ_T(x) = thermal stress distribution]

Superposition Principle

For multiple load cases: K_I_total = K_I_tension + K_I_bending + K_I_pressure + ... [linear superposition within LEFM]

Example — pressure vessel with surface crack: Hoop stress σ_θ = p × r/t → K_I_hoop = σ_θ × √(πa) × F_surface Internal pressure in crack: K_I_pressure = p × √(πa) × F_pressure K_I_total = K_I_hoop + K_I_pressure

Bending + tension: K_I = (σ_membrane × F_m + σ_bending × F_b) × √(π × a) F_m, F_b = different correction factors for membrane vs. bending stress

K-G (SIF — Strain Energy Release Rate) Relationship

Mode I (plane stress): G_I = K_I² / E [J/m²; G_I = energy release rate per unit crack area advance]

Mode I (plane strain): G_I = K_I² × (1-ν²) / E

Mixed mode: G_total = G_I + G_II + G_III [energy release rates from each mode add] G_total = (K_I² + K_II²) × (1-ν²)/E + K_III² × (1+ν)/E [plane strain]

Fracture criterion: K_I = K_Ic (Mode I dominant); G = G_c (critical energy release rate) = K_Ic²(1-ν²)/E

Fracture Toughness (K_Ic) — ASTM E399

Test Method

Specimen geometries: CT (compact tension) or SENB; notched and fatigue pre-cracked Size requirements for valid K_Ic (plane strain): B ≥ 2.5 × (K_Q/σ_y)² [B = specimen thickness] a ≥ 2.5 × (K_Q/σ_y)² [crack length] W - a ≥ 2.5 × (K_Q/σ_y)² [remaining ligament] If all satisfied: K_Q = K_Ic (valid plane strain fracture toughness)

Typical K_Ic values:

Material	K_Ic [MPa√m]	Notes
4340 Steel (σ_y = 1,400 MPa)	50–70	High strength; low toughness
4340 Steel (σ_y = 1,100 MPa)	100–130	Tempered; better toughness
A36 Steel	150–200	Structural; high toughness
7075-T6 Al	24–30	High strength; low toughness
2024-T3 Al	44	Better toughness; aircraft
Titanium 6-4	55–80	Good balance
INCONEL 718	100–110	Aerospace
PMMA (acrylic)	1–2	Brittle polymer
SiC ceramic	2–5	Very brittle

CTOD and J-Integral (Elastic-Plastic)

When LEFM invalid (plastic zone too large): use CTOD or J-integral (ASTM E1820)

LEFM valid if: plastic zone r_p << crack size r_p = (1/6π) × (K_I/σ_y)² [Irwin plastic zone radius; plane strain] Validity: a >> r_p → a > 25 × (K_Ic/σ_y)² [usually satisfied for high-strength materials]

J-integral: J = -∂U_total/∂A [U_total = total potential energy; A = crack area] J = K_Ic²/E (plane stress) or K_Ic²(1-ν²)/E (plane strain) — equivalence at limit

CTOD (Crack Tip Opening Displacement): δ = K_I²/(E × σ_y) × (plane stress) or δ = K_I²/(E × σ_y × (1+ν)) (plane strain) Or from J: δ = J / (m × σ_y) [m = dimensionless factor ≈ 1.0 (plane stress) to 2.0 (plane strain)]

Mixed-Mode Fracture

Erdogan-Sih (Maximum Tangential Stress) Criterion

Crack propagates in direction of maximum σ_θ: σ_θ_max at angle θ_c: θ_c = arctan((K_I - √(K_I² + 8K_II²)) / (4K_II)) [from Erdogan-Sih 1963]

Mixed-mode fracture condition: K_Ic = cos(θ_c/2) × [K_I × cos²(θ_c/2) - (3/2) × K_II × sin(θ_c)] [interaction locus]

Interaction surface (normalized): (K_I/K_Ic)² + (K_II/K_IIc)² = 1 [conservative elliptic criterion for mixed mode; K_IIc ≈ 0.6 K_Ic]

Standards and References

Standard	Scope
ASTM E399	Test method for linear-elastic plane-strain fracture toughness K_Ic
ASTM E1820	J-integral and CTOD fracture toughness measurement
ASTM E647	Fatigue crack growth rate
NASA/NASGRO	Stress intensity factor handbook and crack growth software
Stress Intensity Factors Handbook (Murakami)	Comprehensive SIF table for 2D/3D geometries
Anderson "Fracture Mechanics" (4th ed.)	Primary textbook

Output

Provide: geometry description (specimen/component type; crack type: edge/surface/central/corner; dimensions: W [mm], B [mm], a [mm], c [mm] if surface; loading: P [kN] or σ [MPa]), applied SIF formula (source: handbook/ASTM/Raju-Newman; F(a/W) correction factor; finite width/surface correction), K_I calculation (K_I = σ × √(πa) × F [MPa√m]; step-by-step; K_II, K_III if mixed mode), comparison with K_Ic (material K_Ic [MPa√m]; safety factor on K: K_Ic/K_I; margin [%]; fracture risk: low/marginal/critical), LEFM validity (r_p [mm] = (K_I/σ_y)²/(6π); compare with a and B; plane strain condition: B > 2.5(K_Ic/σ_y)² satisfied?), J-integral or CTOD (if plastic zone large: J_applied [kJ/m²]; J_Ic [kJ/m²]; comparison), mixed mode (if K_II present: θ_c [°]; effective K; Erdogan-Sih interaction check), fatigue crack growth (if cyclic: ΔK = K_max - K_min; Paris law da/dN = C × ΔK^m; critical crack size from K_max = K_Ic), and applicable standard (ASTM E399/E1820; Murakami handbook equation reference).