Fracture Toughness

Fracture Toughness (structures/materials/fracture-toughness)

Use when the task is the fracture toughness of an aerospace material: applied stress intensity K from stress, crack size, and geometry factor, the failure criterion K >= K_IC, the critical crack size at fast fracture, and the plane-strain validity of the K_IC test specimen, as the materials property that gates damage tolerance sizing.

Domain quick reference

• Plane-strain fracture toughness K_IC: the material property measuring resistance to unstable mode I crack extension under a triaxial plane-strain constraint at the crack tip. Units MPa sqrt(m) or ksi sqrt(in). A valid K_IC is a size-independent lower-bound toughness; thin sections show higher, geometry-dependent toughness.
• Applied stress intensity: K = Y * sigma * sqrt(pi * a), with sigma the remote stress in MPa, a the crack size (half length of a through-crack, depth of an edge crack) in meters, and Y the dimensionless geometry factor (1.0 for a crack in an infinite plate, 1.12 for an edge crack). K has units of stress times sqrt(length): MPa sqrt(m).
• Worked anchor: sigma = 200 MPa, a = 5 mm = 0.005 m, edge crack Y = 1.12 gives K = 1.12 * 200 * sqrt(pi * 0.005) = 28.07 MPa sqrt(m).
• Failure criterion: fast fracture initiates when K >= K_IC. At K_IC = 26 MPa sqrt(m) the anchor crack fails; the applied K of 28.07 MPa sqrt(m) exceeds the toughness.
• Critical crack size: a_c = (K_IC / (Y * sigma)) ** 2 / pi. At K_IC = 30 MPa sqrt(m), sigma = 200 MPa, Y = 1.12 the critical crack is a_c = (30 / 224) ** 2 / pi = 5.71 mm. It scales with 1 / sigma^2: doubling the stress quarters the tolerable crack size.
• Plane-strain validity (ASTM E399 test context): a valid K_IC test requires specimen thickness B and crack size a both >= 2.5 * (K_IC / sigma_ys) ** 2, with sigma_ys the 0.2 percent offset yield strength. Anchor: K_IC = 30 MPa sqrt(m), sigma_ys = 500 MPa gives 2.5 * (30 / 500) ** 2 = 9 mm; thinner specimens measure a plane-stress or transitional toughness, not K_IC.
• Damage tolerance link: K_IC is the material property that gates damage tolerance sizing; residual-strength analysis evaluates the remaining strength of the cracked part against K_IC, fatigue growth governs the life between inspections, and MMPDS compiles the statistically based A-basis and B-basis allowables from which design toughness values are taken. The formula and validity rule are common materials-engineering methodology, summary-only.

Workflow

1. Identify the material K_IC and its source: MMPDS allowables or test data for the correct product form, heat treatment, thickness, and loading orientation (LT versus TL), and confirm the section meets the plane-strain validity requirement.
2. Define the crack scenario: crack size a in meters, remote stress sigma in MPa, and the geometry factor Y for the crack configuration (edge crack 1.12, embedded crack 1.0, or the configuration-specific value).
3. Compute the applied stress intensity with stress_intensity: K = Y * sigma * sqrt(pi * a).
4. Check the failure criterion with is_fracture: K >= K_IC means fast fracture initiates at this crack size and stress; otherwise the crack is stable at this load level.
5. Size the critical crack with critical_crack_size: a_c = (K_IC / (Y * sigma)) ** 2 / pi, and compare it with the NDI detectable crack size to set inspection intervals (damage tolerance link).
6. Verify plane strain with plane_strain_valid: thickness and crack size must both exceed 2.5 * (K_IC / sigma_ys) ** 2; if not, the measured toughness is thickness dependent and the analysis needs the thickness-corrected toughness or a full ASTM E399 test program.

Pitfalls

• Confusing K_IC with the Ramberg-Osgood yield point (ramberg-osgood): K_IC is a crack-resistance property in MPa sqrt(m), not a stress; sigma_0.2 is a plain-specimen strength in MPa. High yield strength does not imply high toughness, and many alloys trade the two against each other in heat treatment, so never infer toughness from the stress-strain curve.
• Reading K_IC from the wrong MMPDS table (mmpsd-allowables): MMPDS A-basis and B-basis tables cover static mechanical properties; plane strain fracture toughness values are reported for specific product forms, thickness ranges, and orientations and are not in every allowable table. Match the test condition before using a number.
• Using K_IC where crack growth rate is the gate (crack-growth): K_IC gates the final unstable extension; the life between inspections is governed by da/dN versus delta K (Paris-type growth). Do not size inspection intervals from K_IC alone.
• Confusing critical crack size with residual strength (residual-strength): a_c from K = K_IC is the crack size at final fracture under the applied stress; residual-strength analysis gives the remaining load capability of the cracked part versus crack size, and K_IC is the property that sets that curve at the instability point. They answer different questions about the same cracked part.
• Reaching for fracture toughness when the task is material selection on strength or stiffness (material-selection): material-selection screens on density-normalized strength and modulus; fracture toughness is a separate screening property with its own test validity conditions, and toughness and strength often trade against each other.
• Mixing units: K in MPa sqrt(m) requires sigma in MPa and a in meters. A crack size in millimeters with stress in MPa gives K in MPa sqrt(mm), which is smaller by sqrt(1000) than the MPa sqrt(m) value; convert a to meters first.
• Dropping the geometry factor: an edge crack (Y = 1.12) runs 12 percent higher in K than an embedded crack (Y = 1.0) at the same size; K scales linearly with Y while the critical crack scales with 1 / Y^2.
• Using K_IC outside the plane-strain regime: below the validity dimension 2.5 * (K_IC / sigma_ys) ** 2 the measured toughness rises as the section thins; the published K_IC only applies in the plane-strain regime.
• Ignoring orientation: wrought aerospace alloys are anisotropic, and K_IC in the TL orientation can be well below the LT value; use the orientation that matches the crack path in the part.

Behavior contract (gate 3)

The stress intensity, failure criterion, critical crack size, and plane-strain validity logic is exercised by the gate 3 contract test: scripts/test_fracture_toughness.py against scripts/fracture_toughness_logic.py (stdlib unittest, offline). Run: python3 scripts/test_fracture_toughness.py

Compliance

• Standards referenced, not reproduced: ASTM E399 is named as the test method context for the plane-strain validity rule and MMPDS as the source of material allowables. Both are proprietary standards; this leaf carries only the summary-level formula and validity rule, which are common materials-engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.