Fracture Toughness and Stress Intensity Calculator

Linear Elastic Fracture Mechanics (LEFM) LEFM describes the stress field near a crack tip in a linear elastic material. Developed by George Irwin (USA, 1957), building on Griffith’s energy theory (1921). The stress intensity factor K quantifies the crack tip stress intensity: K = Y × σ × √(π × a) Where: Y = geometry factor (dimensionless, ~1.0 for simple cases) σ = applied stress (MPa) a = crack half-length for internal crack, or crack length for surface crack (m) K is in MPa·√m (or ksi·√in)

Fracture Criterion Fracture occurs when K ≥ K_IC (plane strain fracture toughness). K_IC is a material property measured by standard tests (ASTM E399). Mode I (opening) means normal stress perpendicular to the crack, and it is both the most common case and the most critical.

Typical K_IC Values (MPa·√m) Metals: Steel (high strength, 4340): 50 to 100 | Aluminum 7075-T6: 24 to 31 Titanium Ti-6Al-4V: 44 to 66 | Copper: 30 to 40 | Cast iron: 6 to 20 Ceramics: Alumina (Al₂O₃): 3 to 5 | Silicon nitride: 5 to 9 | Glass: 0.6 to 1.0 Polymers: Polycarbonate: 1.0 to 2.6 | Epoxy: 0.3 to 0.6 | PMMA: 0.7 to 1.6 Composites: CFRP: 20 to 50 | GFRP: 7 to 12

Critical Crack Size a_c = (K_IC / (Y × σ))² / π Any crack larger than a_c will propagate catastrophically. Fracture toughness determines the maximum allowable defect size for a given stress.

Geometry Factors Y Center crack in infinite plate: Y = 1.0 Single edge crack in semi-infinite plate: Y = 1.12 Surface crack, the usual working assumption when the shape is unknown: Y = 1.25 Embedded elliptical (penny-shaped) crack: Y = 0.728 Through crack at the edge of a finite plate: Y = 1.0 to 1.5, depending on the a/W ratio

When LEFM stops being valid, and why it matters

Everything above assumes the material stays elastic except in a small region right at the crack tip. Real metals yield there, and if that plastic zone gets large compared with the crack, the elastic stress field the whole theory is built on no longer describes what is happening. The equations keep returning numbers; the numbers stop being true.

The plastic zone radius under plane strain is:

r_p = (1 / 6π) × (K / σ_y)²

with σ_y the yield strength. Under plane stress, where a thin section can contract freely through its thickness, the zone is three times larger at (1 / 2π) × (K / σ_y)². The usual rule of thumb is that LEFM holds when the crack is at least ten times the plastic zone. Below that you need elastic-plastic fracture mechanics: the J-integral or CTOD (Crack Tip Opening Displacement).

The same ratio controls whether a K_IC value can be measured at all. ASTM E399 requires the specimen thickness, the crack length and the remaining ligament to each be at least:

2.5 × (K_IC / σ_y)²

For a 4340 steel at 1,200 MPa yield with K_IC of 75, that works out at about 10 mm. Thinner than that and the specimen is not in plane strain, so the toughness measured comes out higher than K_IC and applies only to that thickness. This is why thin sheet is genuinely tougher than thick plate of the same alloy, and why substituting a thinner section is not always the conservative choice people assume.

Enter a yield strength below and the calculator checks both conditions for you.