Fatigue Crack Growth

Crack Growth

Stress Intensity, Growth Laws, Integration and Inspection Intervals

What the tool does

A crack of initial size a_i under a cyclic remote stress grows a little on every cycle. Linear elastic fracture mechanics describes the rate with the stress-intensity range ΔK = Y Δσ √(πa) through a growth law da/dN = f(ΔK, R), and the number of cycles from a_i to a final size a_f is N = ∫ da / f(ΔK, R). The final size is either entered or the critical size a_c at which K_max = Y σ_max √(πa) reaches the fracture toughness K_IC (or the plane-stress K_c of the thickness, when that governs). The tool evaluates the geometry factor Y(a) for the chosen configuration, converts the law's constants to a consistent unit system, integrates numerically with Y re-evaluated as the crack grows, and reports the cycles to failure, the crack size at tenths of the life, the a–N and da/dN–ΔK curves, an inspection interval from a detectable crack size, the residual strength against crack size and, optionally, the life under a repeated sequence of constant-amplitude blocks.

Conventions

• Stresses are remote (gross-section) stresses normal to the crack plane; tension positive. ΔK is taken over the full range σ_max − σ_min and R = σ_min/σ_max, so a compressive minimum gives R < 0 and a ΔK larger than K_max. Whether the compressive part does damage is the growth law's business (the Walker exponent or Newman's opening function), not the definition of ΔK.
• a is the half-length of a centre crack, the length of an edge crack, the length from the hole edge of a hole crack and the depth of a surface crack; c is the surface half-length of the surface crack; W is the full plate width (2b in the handbooks); t the thickness; R the hole radius.
• K = Y σ √(πa) everywhere. For the surface crack the handbook form K = σ √(πa/Q) F is converted to Y = F/√Q so that the same integrator serves every geometry; the two tips carry their own Y.
• Stress intensities are shown in ksi·√in (USCS) or MPa·√m (SI); 1 ksi·√in = 1.098843 MPa·√m. The constants are entered in the unit pair of their source and converted on the page.

Geometry factors

• Centre crack, infinite plate: Y = 1 (Irwin).
• Centre crack, finite width: Y = √sec(πa/W), Feddersen's secant, within 0.3 % of Isida's series to 2a/W = 0.7 (Tada, Paris and Irwin). The tool stops the solution at 2a/W = 0.95.
• Single edge crack: Y = 1.122 − 0.231 (a/W) + 10.55 (a/W)² − 21.71 (a/W)³ + 30.382 (a/W)⁴ (Brown and Srawley), 0.5 % for a/W ≤ 0.6, where the tool stops.
• Double edge crack: Y = [1.122 − 0.561 x − 0.205 x² + 0.471 x³ − 0.190 x⁴] / √(1 − x), x = 2a/W (Tada), any x below 1.
• Two symmetric cracks at a hole: Newman's collocation table (NASA TN D-6376, 1971, Table I, uniaxial column) for K = S √(π(R + c)) F(a/R), a/R from 1.01 to 4, interpolated linearly; below 1.01 the factor scales with √c as at a notch root (K_t = 3), above 4 it approaches the Irwin value of a crack of half-length R + c as (R/a)². Bowie's values fitted by Tada agree with the table within 2.5 %. A single crack at a hole is not offered: its factor lies below the two-crack value and tends to √(π(2R + c)/2) for a long crack; enter it as a polynomial if needed.
• Semi-elliptical surface crack: Newman and Raju (NASA TM 83200, 1981): K = S √(πa/Q) F, Q = 1 + 1.464 (a/c)^1.65 (a/c ≤ 1; (c/a)^1.65 above), F = [M₁ + M₂ (a/t)² + M₃ (a/t)⁴] g f_φ f_w with M₁ = 1.13 − 0.09 a/c, M₂ = −0.54 + 0.89/(0.2 + a/c), M₃ = 0.5 − 1/(0.65 + a/c) + 14 (1 − a/c)²⁴, g = 1 + [0.1 + 0.35 (a/t)²](1 − sin φ)², f_φ = [(a/c)² cos²φ + sin²φ]^¼, f_w = √sec(πc/(2b) √(a/t)) for a/c ≤ 1 and the companion set for 1 < a/c ≤ 2; φ = π/2 is the deepest point, φ = 0 the surface. The equation reproduces the finite-element factors of NASA TP-1578 within 5 % for a/t ≤ 0.6 and 7 % at a/t = 0.8 (checked in the test script on 60 table cells). Range: 0.2 ≤ a/c ≤ 2, c/b < 0.5, a/t < 1. The two tips grow with their own ΔK, so the shape changes along the path; the life ends at K_IC or at break-through (a = t), after which the crack is a through crack of length 2c and should be re-analysed as such.
• Entered Y: Y = Y₀ + Y₁ (a/W) + Y₂ (a/W)² + Y₃ (a/W)³ from any handbook; a constant Y (Y₁ = Y₂ = Y₃ = 0) admits the closed form.

Growth laws

• Paris (Paris and Erdogan 1963): da/dN = C ΔK^m. Region 2 only; no R effect, no threshold, no fracture asymptote. A threshold ΔK_th may be entered: the rate is set to zero below it.
• Walker (1970): da/dN = C [ΔK/(1 − R)^(1 − γ)]^m, the form fitted in the FAA database (its m is the γ here). γ = 1 removes the R effect; γ = 0 makes the rate depend on K_max only; aluminum alloys lie near 0.6–0.7 for R ≥ 0. At R = 0 the law is Paris with the same C and m. The FAA fits give a separate exponent for negative R (m−, about 0.02–0.16); the tool applies the one entered.
• Forman (Forman, Kearney and Engle 1967): da/dN = C ΔK^m / [(1 − R) K_c − ΔK]; the rate grows without bound as K_max → K_c, so the curve bends up in region 3. Its C has the units of length / K^m−1. Hudson's constants for 2024-T3 and 7075-T6 sheet are in psi·√in as printed.
• NASGRO (DOT/FAA/AR-05/15 eq. 1): da/dN = C [(1 − f)/(1 − R) ΔK]ⁿ (1 − ΔK_th/ΔK)^p / (1 − K_max/K_c)^q, covering the threshold, the linear and the fracture regions. f = K_op/K_max is Newman's crack-opening function of R, the constraint factor α (1 plane stress, 3 plane strain) and S_max/σ₀: for R ≥ 0, f = max(R, A₀ + A₁R + A₂R² + A₃R³); for −2 ≤ R < 0, f = A₀ + A₁R; A₀ = (0.825 − 0.34α + 0.05α²)[cos(πS_max/2σ₀)]^(1/α), A₁ = (0.415 − 0.071α) S_max/σ₀, A₃ = 2A₀ + A₁ − 1, A₂ = 1 − A₀ − A₁ − A₃. The threshold follows the NASGRO 4 form behind the FAA constants, ΔK_th = ΔK₁ √(a/(a + a₀)) / [(1 − f)/((1 − A₀)(1 − R))]^(1 + R Cth) with Cth = Cth+ for R ≥ 0 and Cth− for R < 0 and the intrinsic crack length a₀ = 0.0015 in; ΔK₁ is the long-crack threshold at R = 0. K_c is the fit's value for the thickness the database names (the thickness correction K_c/K_IC = 1 + B_k exp[−(A_k t/t₀)²] is not applied; enter the K_c of your thickness).

Crack closure and load interaction: the R effect in the Walker and NASGRO laws is the engineering account of plasticity-induced closure. Retardation after an overload (the plastic zone left behind holds the crack open and slows it for some distance) and acceleration after an underload are not modelled: the block-loading option integrates each block at its own constant amplitude, which is conservative where retardation would occur.

Units of C

The coefficient C carries the units of da/dN divided by those of ΔK^m, so its number changes with the unit pair. With L metres per length unit and K Pa·√m per stress-intensity unit, C_SI = C × L / K^m for the Paris, Walker and NASGRO forms and C × L / K^m−1 for Forman's. 1 in = 0.0254 m; 1 ksi·√in = 1.098843 MPa·√m; 1 psi·√in = 1.098843 × 10⁻³ MPa·√m. Barsom's ferrite-pearlite bound 3.6 × 10⁻¹⁰ ΔK³ in/cycle (ksi·√in) becomes 6.9 × 10⁻¹² ΔK³ m/cycle (MPa·√m). The page prints the conversion with the substituted factors. The coefficient is entered as a mantissa and a power of ten (1.67 and −9 for 1.67 × 10⁻⁹).

Critical crack size and threshold

a_c solves Y(a) σ_max √(πa) = K_IC by bisection with the geometry factor re-evaluated; when K_IC is not reached inside the range of the geometry solution the life is given to that limit and a note says so. If K_max at the initial crack already exceeds K_IC the part fails on the first cycle. The threshold check compares ΔK at the initial crack with ΔK_th (entered, or the NASGRO value at the initial size and R): below it the crack does not grow and the life is unbounded.

Integration

The crack size is advanced in N by a fourth-order Runge–Kutta scheme with step doubling (relative tolerance 10⁻⁷ in a) and Y re-evaluated at every stage; the step shrinks to land on the final size, and a step that runs into fracture is cut back until the tip sits at K_IC. For a constant Y and a power law the closed form N = [a_f^1−m/2 − a_i^1−m/2] / [C (Y Δσ √π)^m (1 − m/2)] (ln(a_f/a_i) / [C (Y Δσ √π)²] for m = 2) is written out with the substituted numbers and agrees with the integration to a few parts in 10⁵; for the other geometries a composite Simpson integration of dN/da on a logarithmic grid in a is printed as a check. Most of the life is spent while the crack is small and ΔK low, which the table of crack size at tenths of the life shows.

Inspection interval and residual strength

Damage-tolerance practice sets the inspection interval so that a crack just missed at one inspection (at the detectable size a_d of the method) is found at the next before it reaches a_c: interval = N(a_d → a_c) / 2, two opportunities to detect it. The tool reads N(a_d) off the integrated path. Safety factors on the crack-growth life (2 to 3 is usual), the scatter of the constants and the probability of detection are the analyst's to add. The residual strength σ_c = K_IC / (Y(a) √(πa)) is the remote stress the cracked part can still carry; it falls as the crack grows and equals σ_max at the critical size.

Material constants

The material list fills the fields with printed, open constants: the FAA fatigue crack growth database (DOT/FAA/AR-05/15, 2005) gives NASGRO 4 and Walker fits for thirteen 2000- and 7000-series aluminum alloys in in/cycle and ksi·√in, with K_1c, K_c at the stated thickness and ΔK₁; Hudson (NASA TN D-5390, 1969) gives Forman constants for 2024-T3 and 7075-T6 sheet in psi·√in; Barsom and Rolfe give conservative upper bounds for ferrite-pearlite, martensitic and austenitic steels (Paris form, no toughness, which the analyst enters). The Paris constants of an aluminum entry are the Walker C and m, which the Walker law reduces to at R = 0. The sources are filed in References/Crack-Growth/.

Limits

• Linear elastic fracture mechanics: the plastic zone must be small against the crack and the ligament; the net-section stress should stay below yield.
• Constant-amplitude laws with no load-interaction model; the block option has no retardation.
• Remote tension only (no bending factors, no pin loading, no residual stress).
• Geometry solutions end at their validity limits (2a/W = 0.95, a/W = 0.6 for the single edge crack, a/t = 1 for the surface crack); beyond them the life is not extended.
• The FAA Walker fits hold in the linear region; their K_c and K_1c are the database's values, not a design allowable.