Materials Fatigue Fracture Mechanics From Jcost Fatigue Fracture Cert

The certification

Fatigue failure is what happens when repeated loading, not a single overload, breaks a material. A paper clip bent back and forth snaps not because one bend was too strong but because each cycle deposits a little damage. In materials science this accumulation is described by the Paris-Erdogan law, which links the crack growth rate per cycle, da/dN, to the stress intensity range ΔK through a power law with exponent m. For metals, m typically sits between 2 and 4.

The Recognition Science framework models the same process with its own vocabulary. Each load cycle deposits a recognition cost, written J(Δσ/σ_yield), where Δσ is the stress range and σ_yield is the material's yield strength. When the cumulative cost crosses a canonical band, J(φ), a crack initiates. The framework's machine-checked library of formal theorems packages this into a structure called FatigueFractureCert, which bundles two facts: there are exactly five canonical failure modes, and the damage threshold is the canonical certificate.

The five modes are ductile fracture, brittle fracture, fatigue, creep, and stress corrosion cracking. The certificate proves, by direct computation, that this list has exactly five entries. That count is not arbitrary in the framework: it matches a structural dimension, configDim D = 5, that the framework derives from its forcing chain. The damage threshold is the same canonical certificate used elsewhere in the framework, the one built from the cost function J(x) = (x + 1/x)/2 - 1.

What the certificate does not do is predict a specific Paris exponent. The framework offers a rough estimate, m ≈ 2/φ ≈ 1.24, which sits below the lower end of the measured 2 to 4 range for metals. That estimate is a hypothesis, not a theorem. The certificate itself only establishes the mode count and the threshold structure; it does not validate the Paris law against experimental data, and it does not claim that real materials fail in exactly five ways. The five modes are a modeling choice, a useful taxonomy, not a proven law of nature.

theorem failureModeCount : Fintype.card FailureMode = 5 := by decide

structure FatigueFractureCert where
  five_modes : Fintype.card FailureMode = 5
  damage_threshold : CanonicalCert

inductive FailureMode where
  | ductileFracture | brittleFracture | fatigue | creep | stressCorrosion
  deriving DecidableEq, Repr, BEq, Fintype

What this page does not claim

The certificate does not prove the Paris-Erdogan law or any specific crack growth rate. The five failure modes are not claimed to be exhaustive for all materials and conditions. The Paris exponent estimate m ≈ 1.24 is not a theorem and does not match the measured range.

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

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