Materials Fatigue Fracture Mechanics From Jcost Failure Mode Count

The five failure modes

Fatigue and fracture mechanics is the study of how solid materials break under repeated or sustained stress. A metal part in an airplane wing, a turbine blade, or a bridge experiences millions of load cycles, each one too small to break it outright, until one day a crack appears and grows. The classical Paris-Erdogan law, from 1963, describes how fast that crack grows: da/dN = C × (ΔK)^m, where a is crack length, N is the number of cycles, and ΔK is the stress intensity range. The exponent m is not a universal constant; for metals it typically falls between 2 and 4.

In the Recognition Science framework, each load cycle deposits a recognition cost, a discrete record of the damage accumulated by the material. The framework's central cost function J(x) = (x + 1/x)/2 - 1 measures that deposit. When the cumulative cost crosses a canonical threshold, the framework says a crack initiates. The framework also identifies five canonical failure modes: ductile fracture, brittle fracture, fatigue, creep, and stress corrosion cracking. These five are the subject of a machine-checked theorem in the framework's library of formal theorems.

The theorem, named failureModeCount, establishes that the number of these canonical failure modes is exactly five. It is a counting result, proved by checking each of the five named modes and confirming there are no others in the framework's definition. The proof is computationally trivial, but the claim is not: it says the framework's model of material failure recognizes these five distinct categories and no more. The same library also packages this count with a damage threshold certificate, a formal record that the five-mode count and the threshold condition are both in place.

What the theorem does not claim is just as important. It does not say that real materials fail in exactly five ways. It does not predict which mode will occur under given conditions, nor does it rank the modes by likelihood. It does not derive the Paris-Erdogan exponent m; the framework's suggestion that m ≈ 2/φ ≈ 1.24 for metals is a prediction, not a proved result, and it sits at the edge of the measured 2 to 4 range. The theorem only counts the modes as the framework defines them.

The practical consequence is modest but real. If you work with the framework's model of fatigue, you know the full list of failure categories you must account for. You are not missing a sixth mode that the definitions forgot. That is a small, exact statement, and it is the kind of thing a machine-checked proof is good for: not sweeping physical insight, but a precise boundary on what the model contains.

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

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

What this page does not claim

The theorem does not say real materials fail in exactly five ways. The theorem does not predict which failure mode will occur under given conditions. The framework's value for the Paris-Erdogan exponent m is a prediction, not a proved result.

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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