Materials Fracture Toughness From Jcost Fracture Regime

Fracture regimes

Fracture toughness, written K_IC, is a material's resistance to crack growth: the stress intensity at which an existing crack starts to run. In the Recognition Science framework, this threshold is not a free parameter but a consequence of a forced cost function J(r), where r is the ratio of applied stress intensity to the material's toughness. The framework models the ratio r = K_applied / K_IC as the input to J, and the resulting behavior falls into three bands: below r = 1/φ (where φ ≈ 1.618, the golden ratio) the cost stays low and no crack propagates; as r enters the band around J(φ), sub-critical crack growth begins; above r = 1, the cost rises without bound and fast fracture follows. This is a recognition ledger, a discrete record of events, applied to a continuous mechanical quantity.

The declaration FractureRegime itself is a simple classification: an inductive type with exactly five constructors, named elastic, plastic, creep, fatigue, and envAssisted (environmentally-assisted). The machine-checked library of formal theorems proves that this type has exactly five elements, via the theorem fractureRegimeCount. A certificate structure, FractureToughnessCert, bundles this five-regime count together with a canonical threshold certificate, and a definition provides the certificate. The five regimes match the standard engineering taxonomy of failure modes, and the count is the configurational dimension D = 5 in the framework's language.

The framework also links the Paris-Erdogan law, the empirical relation for crack growth rate da/dN proportional to ΔK^m, to its phi-ladder of exponents. The docstring claims m ≈ 4 for structural metals (rung 4) and m ≈ 2 for ceramics (rung 2), with adjacent material-class exponents ratioing by φ ≈ 1.618. This is a stated relationship in the source material, not a proved theorem in the pack.

What the declaration does not claim: it does not derive the value of K_IC for any specific material from first principles. The threshold is a certificate, an input structure, not an output of the framework. The five-regime count is proved, but the physical interpretation of each regime as a distinct failure mode is a modeling choice, not a theorem. The Paris-Erdogan exponent values are asserted in documentation, not proved in the Lean code. The framework's own bridge from recognition cost to physical fracture mechanics remains an identification, not a derivation.

inductive FractureRegime where
  | elastic | plastic | creep | fatigue | envAssisted
  deriving DecidableEq, Repr, BEq, Fintype

theorem fractureRegimeCount : Fintype.card FractureRegime = 5 := by decide

theorem fractureRegimeCount : Fintype.card FractureRegime = 5 := by decide

structure FractureToughnessCert where
  five_regimes : Fintype.card FractureRegime = 5
  threshold : CanonicalCert

noncomputable def cert := fractureToughnessCert

What this page does not claim

The value of K_IC for any specific material is derived from first principles. The five regimes are proved to correspond to distinct physical failure mechanisms. The Paris-Erdogan exponent values m ≈ 4 and m ≈ 2 are proved theorems in the library.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/FractureToughnessFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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