Materials Fatigue Fracture Mechanics From Jcost Failure Mode

The five failure modes

Fatigue failure happens when cyclic loading accumulates damage. In materials science, a component under repeated stress, like an aircraft wing or a turbine blade, slowly develops cracks even when the peak stress stays below the material's yield strength. The Recognition Science framework models this process with a specific mechanism: each load cycle deposits a recognition cost, a discrete record of the energy the material spends responding to the stress. When the cumulative cost crosses a canonical threshold, a crack initiates.

The framework's machine-checked library of formal theorems defines exactly five canonical failure modes: ductile fracture, brittle fracture, fatigue, creep, and stress corrosion cracking. The FailureMode declaration establishes this as a complete enumeration, not a suggestion. A theorem in the library establishes that the number of modes is exactly five, and a certificate structure bundles this result together with the damage threshold. The five-mode count is a formal fact within the framework, not an empirical observation.

The framework connects these modes to the Paris-Erdogan law, a classical empirical relation that describes crack growth rate under cyclic loading. The law states that the crack growth rate da/dN equals C times ΔK raised to the power m, where ΔK is the stress intensity factor range. The framework predicts that the exponent m is approximately 2/φ, or about 1.24, for metallic materials. This sits at the lower end of the commonly observed range of 2 to 4, and the framework does not claim to explain the spread across different materials.

In Recognition Science, the five modes emerge from a deeper structure. The framework's forcing chain derives a cost function J(x) = (x + 1/x)/2 - 1 from five plain conditions, and this cost function yields a configurational dimension of 5. The five failure modes are not arbitrary categories; they correspond to this derived dimension. The framework treats the classical Paris-Erdogan law as itself derivable from the same cost structure, with the exponent m ≈ 2/φ as a prediction.

What the declaration does not claim: it does not claim that the five modes are the only ways materials can fail in every circumstance, nor that the Paris exponent prediction has been verified across all metallic systems. The framework's library establishes the five-mode count and the cost-function structure; the comparison against measured Paris exponents is an empirical check, not a theorem. The physical bridge from recognition cost to actual crack initiation remains open in the framework, and the declaration does not close that gap.

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

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

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

What this page does not claim

The five modes are not claimed to be exhaustive for all materials and loading conditions. The Paris exponent prediction is not claimed to be verified across all metallic systems. The physical mechanism linking recognition cost to crack initiation is not claimed to be closed.

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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