Fracture Mechanics Stress Intensity Factor
Fracture Mechanics Stress Intensity Factor
Linear elastic fracture mechanics predicts whether a cracked component will fail not from the nominal stress alone but from how sharply that stress concentrates at the crack tip — captured in a single quantity, the stress intensity factor K, which governs crack growth and fracture in brittle and semi-brittle materials alike. A structure with even a small crack can fail well below its nominal yield strength if K reaches the material's fracture toughness Kc, which is why fracture mechanics is essential for inspecting welds, pressure vessels, and aircraft structures where flaws cannot be entirely ruled out.
The stress intensity factor is K = Y*sigma*sqrt(pi*a), a geometry factor times remote stress times the square root of crack length. where Y_geo is a dimensionless geometry correction factor, sigma_app is the remotely applied stress, and a_crack is the crack (half-)length.
Inputs
Y_geo := 1.12=1.12
sigma_app := 150 MPa=150 MPa
a_crack := 0.002 m=2 mm
The square-root-of-length dependence is the hallmark of linear elastic fracture mechanics — it reflects how stress theoretically diverges near an idealized sharp crack tip, tempered by the actual crack size.
K_stress := Y_geo·sigma_app·sqrt(pi·a_crack)=1.33e+07 kg/m^(1/2)/s^2
Results
A computed K on the order of ten MPa*sqrt(m) is typical for a small surface flaw at moderate stress; this value is then compared directly against the material's fracture toughness Kc — if K exceeds Kc the crack propagates catastrophically rather than growing slowly by fatigue. Because K scales with the square root of crack length, a crack must grow four times longer to double K, which is part of why fracture often appears to occur suddenly after a long, quiet period of slow fatigue crack growth.
Assumptions & limitations
- Linear elastic fracture mechanics — assumes small-scale yielding at the crack tip.
- Geometry factor Y is specific to the crack and component geometry used.
- Plane strain/plane stress condition not distinguished here — affects which Kc to compare against.
- Static loading; cyclic (fatigue) crack growth requires a separate Paris-law analysis.