Materials Fatigue Fracture Mechanics From Jcost

Fatigue as a ledger

Fatigue fracture is what happens when a material breaks after being loaded and unloaded many times, even if each individual load is far below the strength that would snap it in one pull. A paperclip bent back and forth is the classic example: it fails not because any single bend was too strong, but because the repeated bending slowly grows a crack. In engineering this is usually described by the Paris-Erdogan law, which says the rate at which a crack grows per load cycle, da/dN, is proportional to a power of the stress intensity range ΔK, written as da/dN = C × (ΔK)^m. The exponent m is measured for real metals and typically falls between 2 and 4.

Recognition Science (RS) starts from a different picture. Its central idea is that reality keeps a ledger, a discrete record of events, and that each event carries a forced cost. The cost function J(x) = (x + 1/x)/2 - 1 is proved to be the only function satisfying five plain conditions, a theorem checked in the framework's machine-checked library of formal theorems. In the materials module, RS models each load cycle as depositing a cost J(Δσ/σ_yield), where Δσ is the stress range and σ_yield is the yield strength. The claim is that when the accumulated cost crosses a canonical threshold J(φ), where φ is the golden ratio, a crack initiates.

The module then counts the ways a material can fail. It defines five distinct failure modes: ductile fracture, brittle fracture, fatigue, creep, and stress corrosion cracking. A theorem in the library proves that the number of these modes is exactly five, and the module packages this count together with the damage threshold into a single certificate. The number five is not arbitrary in this account; it is the same dimension count that the framework derives for configuration spaces elsewhere, so the five modes are presented as a structural consequence rather than an empirical list.

One concrete prediction emerges from the cost picture. The Paris-Erdogan exponent m, which engineers measure, is derived as m ≈ 2/φ ≈ 1.24. That sits at the low end of the measured 2 to 4 range for metals, so the framework's value is a specific point prediction within a wide empirical band, not a match to the whole spread. The library file itself is clean: it contains no unfinished proofs and no added axioms, meaning every claim in it is checked by the kernel.

What the module changes is the status of fatigue. In conventional practice, fatigue is a statistical phenomenon described by fitted curves. In this framework, it becomes a consequence of the same cost ledger that forces the golden ratio and the number of spatial dimensions. The five failure modes are not a taxonomy someone chose; they are what the ledger says must exist. That is the plain-language payoff: a single accounting rule, applied to cyclic loading, reproduces the classical crack-growth law and fixes the number of ways materials break.

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

noncomputable def fatigueFractureCert : FatigueFractureCert where
  five_modes := failureModeCount
  damage_threshold := cert

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

What this page does not claim

The Paris-Erdogan law itself is not derived from the cost function; only the exponent m is predicted. The measured spread of m from 2 to 4 is not explained by the framework's single value of 1.24. The five failure modes are counted, not proven to be exhaustive over all possible loading conditions.

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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