Materials Fracture Toughness From Jcost

Fracture toughness from the cost function

Fracture toughness, written K_IC, is the standard measure of a material's resistance to crack propagation. It is the critical value of the stress intensity factor K, the quantity that describes the magnitude of the stress field near the tip of a crack. When the applied K is below K_IC, a crack is stable; when it reaches K_IC, the crack begins to grow. The ratio r = K_applied / K_IC therefore tracks how close a loaded component is to failure.

The classical Paris-Erdogan law describes how cracks grow under repeated loading. It states that the crack growth rate per cycle, da/dN, is proportional to a power of the stress intensity range, ΔK^m. For structural metals, the exponent m is approximately 4; for ceramics, it is approximately 2. The law is empirical, fitted to decades of fatigue testing.

In Recognition Science, the framework models this threshold using its central cost function, a forced measure of the price of a recognition event. The framework's library of machine-checked formal theorems derives the cost function J(r) = (r + 1/r)/2 - 1 from five plain conditions. Applied to fracture, the framework models the stress intensity ratio r as the argument to J. The threshold behavior falls out: below r = 1/φ, where φ is the golden ratio, J(r) stays below J(φ) and no crack propagates; at the threshold, sub-critical growth begins; above r = 1, J(r) rises and fast fracture follows.

This gives a structure to the Paris-Erdogan exponent. The framework's library establishes that the exponent m follows a phi-ladder, with m ≈ 4 for structural metals and m ≈ 2 for ceramics, adjacent material classes separated by a factor of φ ≈ 1.618. The framework also models five canonical fracture regimes, elastic, plastic, creep, fatigue, and environmentally-assisted, as a single finite type whose cardinality is established to be 5. The module assembles these into a certificate: a structure that packages the five-regime count together with the threshold certification, carrying no unproved axioms.

The practical consequence is a unified picture of crack behavior. Instead of separate empirical laws for each material class, the framework derives the threshold and the exponent ladder from one forced cost function. The classical Paris-Erdogan law remains a measured empirical relation; the framework offers a derivation of its structure, not a replacement of its data.

noncomputable def cert := fractureToughnessCert

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

noncomputable def cert := fractureToughnessCert

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

What this page does not claim

The Paris-Erdogan law itself is not derived; it remains a measured empirical relation. The numerical values of the exponents m ≈ 4 and m ≈ 2 are not established to be exact; they are modeled. The framework does not claim that the five regimes are the only possible fracture mechanisms.

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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