Crack Tip Plasticity Correction

Crack-Tip Plasticity Correction (structures/materials/crack-tip-plasticity-correction)

Use when the task is the small-scale-yielding correction to a linear- elastic crack-tip stress field: the classical Irwin single-pass plastic-zone radius in plane stress and plane strain, the effective crack length that the correction implies, the corrected stress intensity K_eff at that effective crack, the exact Dugdale strip-yield zone for a center crack and its small-scale-yielding asymptote, and the size-rule verdict that judges when the underlying linear-elastic result still applies. It pairs with structures/materials/fracture-toughness for the K_IC frame this correction feeds and with structures/damage-tolerance/crack-growth and structures/damage-tolerance/walker-forman-crack-growth, whose rate laws assume small-scale yielding and use this leaf when that assumption is in question.

Domain quick reference

• Elastic stress intensity (shared LEFM input): K = Ysigmasqrt(pi*a), with sigma the remote stress in MPa, a the crack size in meters, and Y the dimensionless geometry factor.
• Irwin plastic-zone radius: plane stress r_p = (1/pi)(K/sigma_ys)^2; plane strain r_p = (1/(3pi))*(K/sigma_ys)^2, exactly one third of the plane-stress zone at equal K and sigma_ys, reflecting the triaxial constraint at the crack tip in a thick section.
• Single-pass effective crack: a_eff = a + r_p, with r_p evaluated from the uncorrected elastic K (classical Irwin treatment, one pass, no iteration).
• Corrected stress intensity: K_eff = Ysigmasqrt(pia_eff); the k-eff-correction ratio is exactly K_eff/K = sqrt(a_eff/a) at fixed Ysigma.
• Dugdale strip-yield zone (exact, infinite-sheet center crack): rho = a*(sec(pisigma/(2sigma_0)) - 1), with sigma_0 the strip (flow) stress; the zone diverges as sigma approaches sigma_0. Its small-scale-yielding asymptote is rho_ssy = (pi/8)*(K/sigma_0)^2, and the zone coefficient ratio rho_ssy over the Irwin plane-stress coefficient is exactly pi^2/8.
• LEFM-validity size rule (applied-K arithmetic): the elastic result is valid when a is at least 2.5*(K/sigma_ys)^2, equivalently when the plane-stress zone fraction r_p/a is at or below 1/(2.5*pi), about 0.1273.
• Units: sigma, sigma_ys, sigma_0 in MPa; a, r_p, rho, a_eff in meters; K, K_eff in MPa*sqrt(m). Scope: mode I crack, single crack size, no iteration on the Irwin pass; the Dugdale arm applies to a center crack with Y = 1 and applied stress below the strip stress.

Workflow

1. Fix the crack scenario: remote stress sigma, crack size a, geometry factor Y, and the material yield strength sigma_ys, and compute the uncorrected elastic K with stress_intensity.
2. Evaluate the irwin-plastic-zone radius in both constraint states with irwin_plastic_zone, using K from step 1: the plane-stress zone and the reduced plane-strain zone.
3. Form the effective-crack-length with effective_crack_length (a_eff = a + r_p) and the corrected stress intensity K_eff with irwin_effective_correction, which returns k, r_p, a_eff, k_eff and the k-eff-correction ratio k_eff_over_k in one call for a chosen constraint state.
4. Run the small-scale-yielding-check with sxy_validity at the elastic K: it returns the required crack size from the size rule, the a-over-required margin, both zone-to-crack fractions and the LEFM-validity verdict.
5. For a center crack, apply the dugdale-strip-yield-model with dugdale_strip_zone for the exact secant zone, dugdale_ssy_zone for its small-scale-yielding asymptote, and dugdale_effective_correction for the full effective-crack chain (k, rho, a_eff, k_eff, k_eff_over_k) in one call.
6. Compare the exact Dugdale zone against its asymptote to judge how far the load sits beyond the small-scale-yielding limit; confirm the deterministic checks with the contract test scripts/test_crack_tip_plasticity_correction.py.

Worked example

7075-T6 aluminium, sigma_ys = 503 MPa (an MMPDS-typical published value, referenced not reproduced).

Case 1: 5 mm edge crack, Y = 1.12, sigma = 180 MPa, a = 0.005 m.

• K = 1.12180sqrt(pi0.005) = 25.266813 MPasqrt(m).
• Plane-stress zone r_p = 8.031841e-04 m (0.803 mm); plane-strain zone 2.677280e-04 m (0.268 mm), exactly one third.
• Plane-stress correction: a_eff = 5.803184 mm, K_eff = 27.220659 MPasqrt(m), k_eff_over_k = 1.077329 (elastic K under-predicts by 7.7 percent). Plane-strain correction: a_eff = 5.267728 mm, K_eff = 25.934456 MPasqrt(m), k_eff_over_k = 1.026424 (2.6 percent).
• Small-scale-yielding verdict: required_a = 6.308193 mm against the 5 mm crack, a_over_required = 0.792620, so the verdict is LIMIT EXCEEDED: the plane-stress zone fraction 0.160637 exceeds the 0.1273 bound.
• Companion at sigma = 120 MPa (same crack): K = 16.844542 MPa*sqrt(m), k_eff_over_k = 1.035082 (3.5 percent), required_a = 2.803641 mm, verdict VALID: the textbook small-correction band.

Case 2: Dugdale strip-yield on a center crack at 90 percent of the flow stress, Y = 1.0, a = 0.010 m, sigma_0 = 503 MPa, sigma = 452.7 MPa.

• K = 452.7sqrt(pi0.010) = 80.238986 MPa*sqrt(m).
• Exact strip-yield zone rho = 5.392453e-02 m (53.925 mm), 5.4 times the half-crack; effective crack a_eff = 63.924532 mm, K_eff = 202.870645 MPa*sqrt(m), k_eff_over_k = 2.528330 (2.53 times the elastic K).
• SSY asymptote rho_ssy = 9.992974e-03 m (9.993 mm); the exact zone is 5.396 times the asymptote, so the asymptote under-predicts badly at 90 percent of the flow stress and the small-scale-yielding limit is far exceeded.

Verification

• Confirm irwin_plastic_zone at the case 1 K in plane strain equals the plane-stress value divided by exactly 3.
• Confirm irwin_effective_correction's k_eff_over_k equals sqrt(a_eff/a) to roundoff for both constraint states, and that at sigma_ys = 1e9 MPa the ratio collapses to 1 within 1e-12 (the small-scale-yielding limit recovery).
• Confirm dugdale_ssy_zone over irwin_plastic_zone (plane stress) at equal K equals pi^2/8 exactly to roundoff.
• Confirm dugdale_strip_zone grows without bound as sigma approaches sigma_0 (the 0.999sigma_0 zone exceeds the 0.9sigma_0 zone by more than 10x) and raises ValueError at and above sigma_0.
• Confirm sxy_validity's required_a_m equals 2.5pir_p_ps (plane stress) to roundoff, and that the case 1 / case 1b / case 2 verdicts match the worked example (False, True, False).
• Confirm every non-positive stress, crack size, geometry factor, yield strength, strip stress, and every unrecognized constraint string raises ValueError.
• Run the contract test offline: python3 scripts/test_crack_tip_plasticity_correction.py (34 tests, deterministic).

Pitfalls

• Iterating the Irwin correction: this leaf is the classical single-pass treatment, r_p from the uncorrected elastic K, a_eff = a + r_p once. An iterative effective-crack loop is a stated extension, not this contract, and will not match the worked-example values above.
• Reading the plane-stress zone where the section is thick: the plane-strain zone is exactly one third of the plane-stress zone at equal K and sigma_ys; using the plane-stress value in a thick-section, constrained geometry overstates the correction by 3x.
• Treating the Dugdale small-scale-yielding asymptote as exact near the flow stress: rho_ssy under-predicts the exact secant zone by a growing factor as sigma approaches sigma_0 (5.4x at 90 percent of the flow stress in case 2); use dugdale_strip_zone directly once the load leaves the small-scale-yielding band.
• Confusing this leaf's size-rule arithmetic with the K_IC test- specimen validity check (structures/materials/fracture-toughness): the 2.5*(K/sigma_ys)^2 form here judges whether an applied-K elastic result is trustworthy at a given crack size and load; the sibling leaf uses the same arithmetic on K_IC and specimen thickness to judge whether a fracture-toughness test is a valid plane-strain measurement. They share arithmetic, not a claim.
• Feeding the Dugdale strip stress the yield strength without checking the model: the strip (flow) stress sigma_0 is typically taken near the yield strength but is a model choice; dugdale_strip_zone raises ValueError once sigma reaches or exceeds sigma_0, since the zone is undefined at and beyond full-strip yield.
• Reporting Ysigmasqrt(pi*a) as the delivered result: this leaf uses the elastic K only as the input to the zone and effective-crack chain; the rate-law leaves (crack-growth, walker-forman-crack-growth) own the Paris-law and stress-ratio projections that consume K directly.

Related leaves

• structures/materials/fracture-toughness: owns the K_IC frame, the K >= K_IC failure criterion and the ASTM E399 test-specimen plane-strain validity check; this leaf reuses only the applied-K small-scale-yielding size rule as a distinct verdict.
• structures/damage-tolerance/crack-growth: the R = 0 Paris-law growth rate that assumes linear-elastic small-scale-yielding conditions this leaf's verdict gates.
• structures/damage-tolerance/walker-forman-crack-growth: the stress- ratio and K_c-limited growth rates that carry the same small-scale- yielding scope this leaf checks.
• structures/materials/ramberg-osgood: the elastic-plastic stress- strain curve for the material, a separate constitutive model from the crack-tip zone geometry here.

Behavior contract (gate 3)

The contract test scripts/test_crack_tip_plasticity_correction.py (34 methods, stdlib unittest, offline, deterministic) verifies: the case 1, case 1b and case 2 worked-example values for stress_intensity, irwin_plastic_zone, effective_crack_length, irwin_effective_correction, dugdale_strip_zone, dugdale_ssy_zone, dugdale_effective_correction and sxy_validity within tolerance; the plane-strain-over-plane-stress one- third identity and the Dugdale SSY-over-Irwin plane-stress zone coefficient pi^2/8 to roundoff; the k-eff-correction sqrt(a_eff/a) identity on both chains; the small-scale-yielding limit recovery at sigma_ys = 1e9 MPa; the size-rule identity required_a_m = 2.5pir_p_ps; the Dugdale zone divergence guard and asymptote collapse at 1 percent of the flow stress; the monotonic correction growth with load ratio; and ValueError rejection of every non-physical input across the module. The test passes under both /usr/bin/python3 and the pyenv 3.13 interpreter; no exact-float equality is asserted on computed sums.

Compliance

• Standards referenced, not reproduced: MMPDS is the source of the 7075-T6 yield strength used in the worked example; the Irwin and Dugdale plastic-zone relations above are standard fracture-mechanics methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.