Forman Crack Growth Rate (R-ratio)
Fatigue crack growth per cycle with mean-stress (R-ratio) and fracture-toughness effects — the equation under NASGRO and AFGROW.
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The engineering
The Forman equation extends the Paris power law with two effects Paris ignores: mean stress (through the ratio R = Kmin/Kmax) and the acceleration of growth as Kmax approaches the fracture toughness Kc. The denominator (1−R)·Kc − ΔK shrinks toward zero as the crack tip nears instability, so the predicted rate rises sharply in Region III — exactly what test data show. Raising R at fixed ΔK also shrinks the denominator, capturing the well-known result that tension-tension cycling at high mean stress grows cracks faster.
Units are the classic gotcha: C is only meaningful alongside its basis. This card assumes C gives da/dN in mm/cycle with ΔK and Kc in MPa·√m — if your handbook lists C for in/cycle and ksi·√in, convert before entering (1 ksi·√in = 1.0988 MPa·√m). Sanity checks: the rate must exceed the plain Paris value C·ΔKⁿ/Kc·(1−R)⁻¹-adjusted floor, and watch the Kc/Kmax margin — below about 1.5 you are in Region III and a single overload can end the part.
Modern codes (NASGRO, AFGROW) use the elaborated Forman–Newman–de Koning form with threshold and crack-closure terms, but the original three-constant Forman law remains the standard quick estimate for damage-tolerance sizing when you only have C, n, and Kc from a materials handbook.
Where this math comes from
Royce Forman was a NASA engineer at the Manned Spacecraft Center in Houston when Apollo hardware forced the question: Paris's 1961 power law fit the middle of the crack-growth curve, but pressure-vessel and structure data at high load ratios kept running above it. In 1967 Forman, with V. E. Kearney and R. M. Engle, published a modification in the ASME Journal of Basic Engineering that divided the Paris numerator by (1−R)·Kc − ΔK — one extra material constant, and suddenly both the R-ratio shift and the run-up to fracture fell out of a single expression.
The equation became the seed of the crack-growth codes the aerospace world still runs. Forman led development of NASA/FLAGRO (later NASGRO) at Johnson Space Center, and the Air Force built the same lineage into AFGROW at Wright-Patterson — both manuals still present the Forman law as the baseline before the elaborated Forman–Newman–de Koning equation. Any damage-tolerance analysis on Redstone Arsenal or Marshall hardware today traces back to that 1967 paper.
- 1961Paul ParisProposes da/dN = C·ΔKᵐ, the power-law heart of crack-growth prediction.
- 1967R. G. Forman, V. E. Kearney & R. M. EnglePublish the Forman equation, adding R-ratio and fracture-toughness effects to Paris.
- 1970K. WalkerAlternative R-ratio correction via an effective ΔK — the other common mean-stress fix.
- 1992NASA Johnson Space CenterNASA/FLAGRO (later NASGRO) codifies the Forman–Newman–de Koning extension for fracture control.
- 1996USAF / AFGROW teamAFGROW manual carries the Forman law as a standard crack-growth model for damage-tolerance analysis.
See the full timeline of the math behind every calculator →
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