Materials Fatigue Threshold From Jcost Fatigue Cost Reciprocal Symm

The symmetry of fatigue cost

Materials fatigue is the failure of metals and polymers under repeated loading at stresses far below the static yield point. Engineers chart this with Wöhler (S-N) curves, Basquin's law, and Coffin-Manson relations. The Recognition Science framework models fatigue through a recognition cost, a discrete record of events, on the dimensionless stress ratio r := observed_stress / yield_stress. The cost function J(r) = (r + 1/r)/2 - 1 has its minimum at r = 1, meaning no perturbation from the yield baseline, and rises as r moves away from 1 in either direction.

The declaration fatigueCost_reciprocal_symm establishes that for any positive stress ratio r, the fatigue cost at r equals the fatigue cost at its reciprocal 1/r. This is a direct consequence of the J-cost function's reciprocal symmetry, a property proved in the machine-checked library of formal theorems. In plain terms, the cost of operating at a stress ratio of 0.8 is the same as the cost at 1.25, and the cost at 0.5 equals the cost at 2.0. The symmetry is exact and holds for all positive r, with the cost at r = 1 being zero and the cost positive for any r not equal to 1.

The framework uses this cost function to define an endurance threshold, the fatigue limit below which infinite life is structurally permitted. This threshold is set at the canonical golden-section cost value J(φ), which lies in the band (0.11, 0.13). The framework defines two mutually exclusive regimes: infinite life when the fatigue cost is below the threshold, and fatigue failure when the cost is at or above it. The theorem regimes_exclusive proves these two states cannot both hold for the same stress ratio.

What the symmetry does not claim is important. It does not assert that the fatigue cost function itself is the only possible model of fatigue, nor does it claim that the endurance threshold value of approximately 0.12 is a measured material constant. The threshold band is a derived property of the J-cost function, not an empirical measurement from a specific steel or polymer. The framework's comparison to the common fatigue limit of about 0.5 times yield stress for many steels is an empirical check, not a theorem. The reciprocal symmetry is a proved mathematical fact about the J-cost function, but its application to real materials remains a modeling choice.

theorem fatigueCost_reciprocal_symm {r : ℝ} (hr : 0 < r) :
    fatigueCost r = fatigueCost r⁻¹ := Cost.Jcost_symm hr

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

/-- 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)

/-- 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]

What this page does not claim

The fatigue cost function is the only possible model of material fatigue. The endurance threshold value of approximately 0.12 is a directly measured material constant. The framework derives the empirical fatigue limit of 0.5 times yield stress for steels; it only notes a correspondence.

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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