Irwin Mode I Stress Intensity Factor K_I and Critical Crack Size Formula

Catch the SQRT(1000) millimeter-to-meter unit trap in fracture spreadsheets

Plane-strain fracture toughness K_IC is tabulated in MPa*sqrt(m), whereas ultrasonic or dye-penetrant inspections report crack depth a_mm in millimeters. Feeding raw millimeters into K_I = Y * sig * SQRT(PI() * a) without dividing a_mm by 1,000 inflates K_I by SQRT(1,000) = 31.62x and triggers false condemnation of safe structures!

Verify on an edge-cracked alloy tension plate: K_IC = 55 MPa*sqrt(m) (B2), edge-crack geometry factor Y = 1.12 (B3), tensile stress sig = 250 MPa (B4), and detected crack depth a_mm = 6.0 mm (B5, i.e., 0.006 m). Here K_I = 1.12 * 250 * SQRT(PI() * 0.006) = 280 * 0.137294 = 38.4422 MPa*sqrt(m), fracture safety factor SF_K = 55 / 38.4422 = 1.4307, and critical crack length a_crit = 1,000 * (55 / 280)^2 / PI() = 12.2817 mm (= 6.0 * 1.4307^2).

Mode I Stress Intensity Factor K_I Formula

Convert inspection millimeters B5 / 1000 inside SQRT(PI() * a_m) and guard non-positive dimensions or toughness.

=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0), "Invalid Fracture Input", ROUND(B3 * B4 * SQRT(PI() * (B5 / 1000)), 4))

Critical Crack Size a_crit (mm) & Stress Doubling Check

Because a_crit scales inversely with sig^2, doubling tensile stress in B4 from 250 MPa to 500 MPa slashes critical flaw size fourfold from 12.2817 mm down to 3.0704 mm (pushing the 6 mm flaw into unstable brittle fracture with K_I = 76.8845 > 55!).

=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid Fracture Input", ROUND(1000 * ((B2 / (B3 * B4))^2) / PI(), 4))

Further reading

• Arte Sheets Lesson — Paris Law Fatigue Crack Growth Cycles to Critical Fracture
• Arte Sheets Lesson — Von Mises Equivalent Stress and Tresca Safety Factor