Walker Forman Crack Growth

Walker-Forman Crack Growth (structures/damage-tolerance/walker-forman-crack-growth)

Use when the task is the stress-ratio-affected (mean-stress and toughness-limited) fatigue crack growth rate of a mode I through- thickness crack under constant-amplitude cyclic loading: the Walker equivalent stress intensity range, the Forman rate with the K_c-limited denominator, and the crack extension over a stated cycle block at constant or piecewise stress ratio R. The Walker relation (Walker 1970, ASTM STP 462) and the Forman relation (Forman, Kearney and Engle 1967, ASME Journal of Basic Engineering) correct the da/dN rate of the linear-elastic fracture mechanics domain for the stress ratio, where goodman-diagram type corrections act on the S-N stress-life amplitude domain instead; the R = 0, gamma = 1 limit reproduces the constant- amplitude Paris baseline of the crack-growth sibling, which owns the un-corrected Ysigmasqrt(pi*a) product as a delivered result. This leaf reuses that form only as the input to the range correction. The fracture endpoint stays with residual-strength: here K_c enters only as the material constant inside the Forman rate denominator, and the arm raises a deterministic guard at dK = (1-R)*K_c, the fracture state. Scope: mode I through-thickness crack in a wide panel (geometry factor Y constant), single half-crack length a growing from a0, far-field tension, constant amplitude and constant R within a segment with R in [-1, 1) and the Walker exponent gamma in (0, 1], linear-elastic small-scale-yielding conditions, material constants C, m, C_F, gamma and K_c given (no da/dN test-data fitting). No crack-closure (Elber) corrections, no threshold, no spectrum cycle counting and no variable-amplitude sequence beyond piecewise-constant R segments.

Domain quick reference

• Applied cycle pinned by sigma_max and R = sigma_min/sigma_max = K_min/K_max: sigma_min = Rsigma_max and the range dsigma = sigma_max(1 - R). Negative R is handled by the pinned formulas as written with no R = 0 clamp; both arms stay defined down to R = -1 and the equivalent range falls below dK under a compressive mean stress.
• Mode I stress intensity K = Y * sigma * sqrt(pi * a) in MPasqrt(m), with sigma in MPa and a in meters. K_max = Ysigma_maxsqrt(pia) and the applied range dK(a) = Ydsigmasqrt(pia) = K_max(1 - R).
• Walker equivalent range (Walker 1970, ASTM STP 462): dK_bar = dK/(1 - R)^(1 - gamma); the equivalent closed form dK_bar = Ysigma_maxsqrt(pia)(1 - R)^gamma holds to roundoff. Recovery limits: R = 0 gives dK_bar = dK for any gamma; gamma = 1 gives dK_bar = dK for any R (no correction); at R = -1, dK_bar = dK/2^(1 - gamma) (dK/sqrt(2) at gamma = 0.5, exact).
• Walker rate: da/dN = C * dK_bar^m in m/cycle. Walker's published gamma = 0.5 fits the 2024-T3 and 7075-T6 aluminum data, so dK_bar = dK/sqrt(1 - R) in the worked examples.
• Forman rate (Forman, Kearney, Engle 1967): da/dN = C_F * dK^m / ((1 - R)*K_c - dK). The denominator vanishes at dK = (1 - R)K_c, equivalent to K_max = K_c (dK = K_max(1 - R)), the fracture state; the arm is undefined at and beyond that point and raises ValueError. The terminal acceleration as the denominator approaches zero is the kc-limited growth this leaf exists to capture: the forman-versus-paris ratio (C_F/C)/((1 - R)*K_c - dK) diverges like (1/(1 - R))/(1 - K_max/K_c) as K_max/K_c approaches 1.
• Paris baseline (comparison denominator only, never a growth projection): C*dK^m, the R = 0, K_c-infinite limit of both arms.
• Material constants: C and m are the crack-growth Paris arm constants of the sibling unit convention (anchor C = 1e-11, m = 3, which reproduces the sibling's own 8e-8 m/cycle at dK = 20 when R = 0 and gamma = 1). C_F is the Forman arm's own fitted constant; the worked-example value C_F = 1e-9 equals CK_c numerically with K_c = 100 MPasqrt(m), an example fit whose R = 0 low-range limit collapses onto the Paris arm.
• Units (single convention, inherited from the crack-growth sibling): sigma in MPa, a in meters, K, dK and K_c in MPasqrt(m), C in (m/cycle)(MPasqrt(m))^-m, C_F in (m/cycle)(MPa*sqrt(m))^(1 - m), da/dN in m/cycle.
• FAR 25.571 damage tolerance practice for transport aeroplanes frames the certification context by name only (far-25, cs-25 reference-only in standards-map.yaml, never reproduced).

Workflow

1. Pin the applied cycle and crack: sigma_max, the stress ratio R, the initial half-crack a0 and the geometry factor Y; evaluate the mode I stress intensity K_max and the applied range dK with stress_intensity (K = Ysigmasqrt(pia), dK from dsigma = sigma_max(1 - R)). The module rejects non-positive stress, crack length and geometry factor with ValueError.
2. Apply the walker-equation equivalent range: compute dK_bar = dK/(1 - R)^(1 - gamma) with walker_equivalent_range using the material gamma exponent; confirm the R = 0 and gamma = 1 recovery limits (both give dK_bar = dK) as the no-correction sanity check.
3. Evaluate the R-corrected rate arms: the walker-equation rate from walker_dadN (CdK_bar^m) and the forman-equation rate with the kc-limited denominator from forman_dadN (C_FdK^m/((1-R)K_c - dK)), plus the walker-versus-paris and forman-versus-paris ratios against the zero-R Paris baseline CdK^m with walker_vs_paris_ratio and forman_vs_paris_ratio.
4. Project the block extension: block_extension marches the selected model over the stated cycle block with forward-Euler substeps (1.0 cycle each when cycles <= 5000, else 5000 equal substeps) that re-evaluate dK from the growing crack at every substep, and returns the R-corrected rate table rows at cycles 0, N/4, N/2, 3N/4, N plus the block extension, start and final rates, and dK / dK_bar bookends.
5. Handle a piecewise stress ratio R: piecewise_block_extension applies the ordered (cycles, sigma_max, R) segments, each run by the identical march starting from the previous segment's final crack length; the split identity (two equal segments reproduce one constant-R block bit for bit) is the built-in consistency check.
6. Watch the kc-limited singularity guard: the forman-equation arm raises ValueError when the marching crack reaches dK = (1-R)*K_c (peak K = K_c, the fracture state owned by residual-strength), stating the cycle at which peak K reaches K_c; the walker-equation arm carries no K_c term and marches on.
7. Verify: run the gate 3 behavior contract (python3 scripts/test_walker_forman_crack_growth.py) under both /usr/bin/python3 and the pyenv 3.13.12 interpreter; the closed-form identities and the determinism sha256 of the canonical dump hold under both.

Worked example

2024-T3 aluminium fuselage panel, central through-crack (Y = 1.0), material constants C = 1e-11 (m/cycle)(MPasqrt(m))^-3, m = 3.0, gamma = 0.5, K_c = 100 MPasqrt(m), C_F = 1e-9 (m/cycle)(MPasqrt(m))^(1 - m) (= CK_c numerically, the example fit). All values below are the real outputs of scripts/ walker_forman_crack_growth_logic.py, bitwise identical to the wave-45 prep anchor under both interpreters (canonical dump sha256 791d233f3a5c74e9030d56c2359d50b920605b5bba3777f12a7ba401175dab62).

Case 1 (tensile mean, R = 0.35): sigma_max = 100 MPa (sigma_min = 35 MPa, dsigma = 65 MPa), a0 = 0.02 m, block of 2000 cycles. K_max0 = 25.06628274631 MPasqrt(m) (K_max/K_c = 0.2507), dK0 = 16.2930837851015, Walker equivalent dK_bar0 = 20.209083229248 MPasqrt(m). Rates at block start: zero-R baseline 4.32523663134402e-08, Walker 8.25353196314273e-08, Forman 8.88012825993901e-08 m/cycle, so walker/paris = 1.90822668598782 and forman/paris = 2.05309651628924: both arms nearly double the R = 0 rate at this range. Walker 2000-cycle block: a_final = 0.0201660975842774 m (extension 0.166098 mm), rate growing to 8.3565620180878e-08 m/cycle. Forman block: a_final = 0.0201789260510205 m (extension 0.178926 mm). Far from K_c the two arms differ only by the R-ratio correction shape.

Case 2 (compressive mean, R = -0.4): sigma_max = 140 MPa (sigma_min = -56 MPa, dsigma = 196 MPa), a0 = 0.05 m, block of 2000 cycles. K_max0 = 55.4865821664841 (K_max/K_c = 0.5549), dK0 = 77.6812150330778, Walker equivalent dK_bar0 = 65.6526093976865 MPa*sqrt(m), BELOW dK0: the compressive mean pulls the equivalent range down. Rates: zero-R baseline 4.6875728436968e-06, Walker 2.82980152371443e-06 (walker/paris = 0.603681610520369 = 1.4^(-1.5), below 1) and Forman 7.52192592680504e-06 m/cycle (forman/paris = 1.60465259476863, above 1 because K_c is felt). Walker block: a_final = 0.0561787832844155 m (extension 6.179 mm). Forman block: a_final = 0.0727588016051557 m (extension 22.759 mm), the kc-limited model grows the same block 3.68 times farther as the denominator (1-R)K_c - dK falls from 62.32 to 46.29 MPasqrt(m); the difference IS the damage tolerance life driver.

Near-critical sweep (sigma_max = 100 MPa, R = 0.35): as a grows, the forman/paris ratio equals 100/((1-R)*K_c - dK) = (1/0.65)/(1 - K_max/K_c), reading 2.0531 (a = 0.02), 4.9069 (0.15), 13.5222 (0.25), 52.7104 (0.30) and 117.0871 (0.31, K_max/K_c = 0.98686) while walker/paris stays 1.9082: the terminal acceleration of the kc-limited denominator is the whole difference. The singularity sits at dK = (1-R)K_c = 65 MPasqrt(m), i.e. K_max = K_c, beyond which block_extension raises.

Piecewise-R block (sigma_max = 100 MPa throughout): 1000 cycles at R = 0.35 then 1000 at R = 0.6, a0 = 0.02 m. Walker model a_final = 0.0201229426364184 m with segment extensions 8.2791219901783e-05 and 4.01514165165699e-05 m (raising R cuts dsigma from 65 to 40 MPa and the rate in the second segment falls despite the higher mean stress); Forman model a_final = 0.0201230576354857 m. Split identity: two 1000-cycle segments at R = 0.35 reproduce the single 2000-cycle block bit for bit under both models.

Verification

• Confirm the anchors: stress_intensity(65.0, 0.02, 1.0) = 16.2930837851015, stress_intensity(100.0, 0.02, 1.0) = 25.06628274631; walker_equivalent_range(16.2930837851015, 0.35, 0.5) = 20.209083229248; walker_dadN = 8.25353196314273e-08 and forman_dadN = 8.88012825993901e-08 m/cycle at the case 1 range, all within the spec tolerances (contract test asserts to 1e-9 relative).
• Confirm the recovery limits: R = 0 recovery and gamma = 1 recovery give dK_bar = dK exactly; at R = -1, gamma = 0.5 the relation dK_bar = dK/sqrt(2) holds to roundoff.
• Confirm the closed-form identities: the gamma = 0.5 square identity dK_bar^2*(1 - R) = dK^2 (residual -5.68e-14); the equivalent closed form dK_bar = Ysigma_maxsqrt(pia)(1 - R)^gamma (residual -3.55e-15); walker_vs_paris_ratio = (1 - R)^(m*(gamma - 1)) (residual -2.22e-16, ratio 1.90822668598782); forman_vs_paris_ratio = (C_F/C)/((1 - R)*K_c - dK) (residual 0.0, ratio 2.05309651628924).
• Confirm the direction bound at one fixed range: dK_bar(R = -0.4) = 13.7701690836267 < dK_bar(R = 0) = 16.2930837851015 < dK_bar(R = 0.35) = 20.209083229248 (a tensile mean accelerates and a compressive mean retards the Walker rate against the zero-R baseline).
• Confirm the block outputs of both cases and both models, the station rows of the R-corrected rate table (monotone in a and rate, with the forman/paris ratio climbing 1.60 to 2.16 across the case 2 block while walker/paris stays pinned at 0.6037), the piecewise-R results, the split identity, the single-cycle march consistency (extension equals the rate at the start to a 7.06e-12 relative residual) and the superlinear doubling of the extension with the cycle count.
• Confirm ValueError rejection of every non-physical input class: the function-level domains (stress, crack length, geometry factor, range, R outside [-1, 1), gamma outside (0, 1], C, C_F, K_c, m, dK at or beyond (1 - R)*K_c) and the block-level guards (non-positive or non-integer cycles, unknown model, a forman block starting at or reaching the singularity mid-march with the cycle in the error, empty piecewise segments, a segment at R = 1).
• Confirm determinism: two identical full runs produce identical bits; the canonical dump sha256 791d233f3a5c74e9030d56c2359d50b920605b5bba3777f12a7ba401175dab62 holds under both /usr/bin/python3 (3.9.6) and the pyenv 3.13.12 hook interpreter; the logic module imports only math and has no RNG.
• Run the gate 3 contract test offline under both interpreters in well under 1 s.

Related leaves

• structures/damage-tolerance/crack-growth: the R = 0 constant-amplitude arm of this pack; it owns the Ysigmasqrt(pi*a) stress intensity evaluation as a delivered product and the Paris law rate and cycles-to-critical projection. Its unit convention and Paris anchor (C = 1e-11, m = 3, dK = 20 gives da/dN = 8e-8 m/cycle) are inherited here, and its zero-R rate is the comparison baseline of both arms of this leaf. Questions that lead with a nonzero stress ratio R or a K_c-limited rate belong to this leaf.
• structures/damage-tolerance/residual-strength: owns the fracture endpoint: residual strength, the critical crack length at which the applied stress drives K to K_c, and the fracture margins. K_c enters this leaf only as the material constant inside the Forman rate denominator, and the singularity guard hands the fracture state to that sibling.
• structures/damage-tolerance/widespread-fatigue-damage: the MSD/MED multiple-site damage screening of FAR 25.571(b)/(c), no rate content.
• structures/damage-tolerance/bird-strike: the certification impact energy vein; its residual_strength_fraction is a post-impact knock- down of a struck component, not a growing-crack rate.
• structures/fatigue/goodman-diagram: the S-N infinite-life mean-stress correction (stress-life domain); Goodman, Gerber and Soderberg correct the allowable stress amplitude, while Walker and Forman correct the da/dN rate of the crack-growth (LEFM) domain this leaf lives in.
• structures/fatigue/stress-life-curve and structures/fatigue/strain-life-fatigue: stress-life curve fitting and the strain-life domain under fully reversed loading; no leaf of the fatigue pack touches da/dN.

Pitfalls

• Mixing units: passing sigma or dK in Pa while C is in (m/cycle)(MPasqrt(m))^-m (or the reverse) shifts the rate by (1e6)^m. Keep the single convention: sigma in MPa, a in meters, K, dK and K_c in MPasqrt(m), C in (m/cycle)(MPasqrt(m))^-m and C_F in (m/cycle)(MPa*sqrt(m))^(1 - m).
• Treating R as zero: the whole point of this leaf is the stress-ratio correction. At R = 0.35 both corrected arms nearly double the zero-R baseline; running the uncorrected rate silently drops the mean-stress and K_c-proximity content of the answer.
• Holding dK constant while the crack grows: the block march re-evaluates dK = Ydsigmasqrt(pi*a) from the growing crack at every substep; a constant-range shortcut understates the extension of a fast-growing crack (doubling the cycles more than doubles the extension).
• Running the forman-equation arm at or beyond the singularity: the denominator (1 - R)*K_c - dK vanishes at dK = (1 - R)*K_c, i.e. K_max = K_c, the fracture state. The arm raises ValueError there; the fracture endpoint, critical crack length and margins belong to residual-strength, not to a rate computation.
• Confusing the correction domains: Walker and Forman correct the da/dN rate of the LEFM domain; Goodman, Gerber and Soderberg correct the S-N amplitude of the stress-life domain (goodman-diagram). The stress ratio R is defined the same way (R = sigma_min/sigma_max) in both, but the outputs are not interchangeable.
• Clamping negative R to zero: the pinned formulas handle R in [-1, 1) as written; an R = 0 lower clamp would erase the compressive- mean retardation the Walker arm is built to show (at R = -0.4 the equivalent range falls below dK).
• Forgetting the certification context is reference-only: FAR 25 and CS 25 are named as context only, never reproduced; the Walker and Forman relations are standard published engineering methodology, summary-only per standards-map.yaml.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_walker_forman_crack_growth.py

The test covers, step by step, the stress intensity anchors and the dK = K_max*(1 - R) range split (step 1), the walker-equation equivalent range with the recovery limits, the R = -1 square-root relation, the square identity and the equivalent closed form (step 2), the walker-equation and forman-equation rates and both model-versus-baseline ratios including the R = 0 far limit, linearity in C_F and the monotone near-critical sweep (step 3), the case 1 and case 2 block outputs and the R-corrected rate table rows of both models (step 4), the piecewise-R block and the bit-for-bit split identity (step 5), the ValueError battery of function domains and the kc-limited singularity guard with the cycle in the error (step 6), and the canonical dump determinism sha256 791d233f3a5c74e9030d56c2359d50b920605b5bba3777f12a7ba401175dab62 plus the stdlib-only purity of the logic module (step 7). It passes under both /usr/bin/python3 (3.9.6) and the pyenv 3.13.12 interpreter. 33 tests, deterministic, well under 1 s.

Compliance

• The Walker equivalent range and the Forman rate are published engineering methodology (Walker 1970, ASTM STP 462; Forman, Kearney and Engle 1967, ASME Journal of Basic Engineering), summary-only paraphrase. The airworthiness context is the FAR-25.571 damage tolerance basis of transport aeroplanes and its CS-25 counterpart, referenced by name only per standards-map.yaml (both reference-only, never reproduced).
• compliance: STANDARDS-REF, gated: false.