Materials Fatigue Threshold From Jcost

The fatigue threshold

Materials fatigue is the failure of metals and polymers under cyclic loading at stresses far below the static yield strength. A steel bridge or aircraft wing can snap after millions of load cycles even though each individual stress is well within its elastic limit. Engineers describe this with the Wöhler curve (S-N curve), Basquin's law, and the Coffin-Manson relation, which chart how the number of cycles to failure drops as the stress amplitude rises. A key feature for many steels is the endurance limit, roughly half the yield stress, below which the material can in principle survive an infinite number of cycles.

The classical picture is empirical: the endurance limit is a measured material property, not a number derived from first principles. The Recognition Science framework offers a formal account. It models fatigue using a dimensionless stress ratio r, defined as observed stress divided by yield stress. The framework's cost function J(r) = (r + 1/r)/2 - 1 assigns a positive cost to any deviation from the yield baseline r = 1, where the cost is zero. This per-cycle cost accumulates, so sub-yield operation carries a steadily growing penalty that the framework associates with fatigue damage.

In Recognition Science, the endurance limit corresponds to a specific threshold value of this cost function evaluated at the golden ratio φ. The framework proves that this threshold, EnduranceThreshold = J(φ), lies strictly between 0.11 and 0.13. The module defines two mutually exclusive regimes: infinite life when the cost is below this threshold, and fatigue failure when the cost is at or above it. The theorem regimes_exclusive proves these two conditions cannot both hold for the same stress ratio.

The framework's library, a machine-checked collection of formal theorems, establishes these results with zero unproved assumptions. The fatigue cost function is shown to be zero at yield, symmetric under reciprocal stress ratios (r and 1/r cost the same), and strictly positive for any stress ratio other than 1. The threshold band (0.11, 0.13) is a proved theorem, not an empirical fit. This places materials fatigue alongside other phenomena the framework treats as manifestations of the same cost quantum, including plaque vulnerability, ignition, and magnetic reconnection.

What this means in plain language: the framework does not predict a specific endurance limit for a given steel alloy. It identifies the mathematical structure of the threshold, a universal band on a normalized cost scale, and proves the logical consistency of the two fatigue regimes. The empirical comparison to measured endurance limits remains a separate check, not a theorem. The framework's contribution is a formal, axiom-clean derivation of the threshold's existence and its numerical band.

theorem fatigueCost_zero_at_yield : fatigueCost 1 = 0 := Cost.Jcost_unit0

theorem fatigueCost_pos_off_yield {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < fatigueCost r := Cost.Jcost_pos_of_ne_one r hr hne

/-- Endurance threshold lies in the canonical band. -/
theorem endurance_threshold_band :
    0.11 < EnduranceThreshold ∧ EnduranceThreshold < 0.13 := by
  unfold EnduranceThreshold
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  rw [Cost.Jcost_eq_sq hphi_ne]
  have h_lo : (1.61 : ℝ) < phi := Constants.phi_gt_onePointSixOne
  have h_hi : phi < (1.62 : ℝ) := Constants.phi_lt_onePointSixTwo
  have hpos : (0 : ℝ) < 2 * phi := by
    have : (0 : ℝ) < phi := Constants.phi_pos
    linarith
  refine ⟨?lo, ?hi⟩
  · rw [lt_div_iff₀ hpos]
    nlinarith [h_lo, h_hi]
  · rw [div_lt_iff₀ hpos]
    nlinarith [h_lo, h_hi]

/-- Infinite-life and fatigue-failing are mutually exclusive. -/
theorem regimes_exclusive {r : ℝ} :
    ¬ (IsInfiniteLife r ∧ IsFatigueFailing r) := by
  rintro ⟨h_lt, h_ge⟩
  exact (lt_irrefl _) (lt_of_lt_of_le h_lt h_ge)

What this page does not claim

This answer does not claim that the framework predicts a specific endurance limit for any particular material. This answer does not claim that the framework's cost function is derived from the physics of dislocation motion or crack propagation. This answer does not claim that the framework's threshold band is an empirical measurement.

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

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